Enumeration of racks, quandles and Bruck loops Petr Vojt echovsk y - - PowerPoint PPT Presentation

Enumeration of racks, quandles and Bruck loops

Petr Vojtˇ echovsk´ y

University of Denver

Loops ’19 Budapest University of Technology and Economics, Hungary July 7–13, 2019

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 3 / 66

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 4 / 66

Coauthors

This is joint work with

• Izabella Stuhl (Penn State University),
• Seung Yeop Yang (Kyungpook National University)

Coauthors

This is joint work with

• Izabella Stuhl (Penn State University),
• Seung Yeop Yang (Kyungpook National University)

and also with

• Dylene Agda Souza de Barros (Federal University of Uberlandia),
• Alexander Grishkov (University of Sao Paulo),
• Alexander Hulpke (Colorado State University),
• Pˇ

remysl Jedliˇ cka (Czech University of Life Sciences),

• Michael Kinyon (University of Denver),
• G´

abor Nagy (Budapest University of Technology and Economics),

• David Stanovsk´

y (Charles University).

Basic definitions (since this is the first lecture)

• An algebra (Q, ·, \, /) is a quasigroup if

x · (x\y) = y, x\(x · y) = y, (x · y)/y = x, (x/y) · y = x.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 6 / 66

Basic definitions (since this is the first lecture)

• An algebra (Q, ·, \, /) is a quasigroup if

x · (x\y) = y, x\(x · y) = y, (x · y)/y = x, (x/y) · y = x.

• A quasigroup is a loop if there is 1 ∈ Q such that 1 · x = x · 1 = x.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 6 / 66

Basic definitions (since this is the first lecture)

• An algebra (Q, ·, \, /) is a quasigroup if

x · (x\y) = y, x\(x · y) = y, (x · y)/y = x, (x/y) · y = x.

• A quasigroup is a loop if there is 1 ∈ Q such that 1 · x = x · 1 = x.
• Mlt(Q) = Lx, Rx : x ∈ Q is the mutiplication group,

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 6 / 66

Basic definitions (since this is the first lecture)

• An algebra (Q, ·, \, /) is a quasigroup if

x · (x\y) = y, x\(x · y) = y, (x · y)/y = x, (x/y) · y = x.

• A quasigroup is a loop if there is 1 ∈ Q such that 1 · x = x · 1 = x.
• Mlt(Q) = Lx, Rx : x ∈ Q is the mutiplication group,
• Mltℓ(Q) = Lx : x ∈ Q is the left multiplication group,

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 6 / 66

Basic definitions (since this is the first lecture)

• An algebra (Q, ·, \, /) is a quasigroup if

x · (x\y) = y, x\(x · y) = y, (x · y)/y = x, (x/y) · y = x.

• A quasigroup is a loop if there is 1 ∈ Q such that 1 · x = x · 1 = x.
• Mlt(Q) = Lx, Rx : x ∈ Q is the mutiplication group,
• Mltℓ(Q) = Lx : x ∈ Q is the left multiplication group,
• Inn(Q) = {f ∈ Mlt(Q) : f (1) = 1} is the inner mapping group,

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 6 / 66

Basic definitions (since this is the first lecture)

• An algebra (Q, ·, \, /) is a quasigroup if

x · (x\y) = y, x\(x · y) = y, (x · y)/y = x, (x/y) · y = x.

• A quasigroup is a loop if there is 1 ∈ Q such that 1 · x = x · 1 = x.
• Mlt(Q) = Lx, Rx : x ∈ Q is the mutiplication group,
• Mltℓ(Q) = Lx : x ∈ Q is the left multiplication group,
• Inn(Q) = {f ∈ Mlt(Q) : f (1) = 1} is the inner mapping group,
• Nuc(Q) = {x ∈ Q : x associates with all y, z ∈ Q} is the nucleus,

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 6 / 66

Basic definitions (since this is the first lecture)

• An algebra (Q, ·, \, /) is a quasigroup if

x · (x\y) = y, x\(x · y) = y, (x · y)/y = x, (x/y) · y = x.

• A quasigroup is a loop if there is 1 ∈ Q such that 1 · x = x · 1 = x.
• Mlt(Q) = Lx, Rx : x ∈ Q is the mutiplication group,
• Mltℓ(Q) = Lx : x ∈ Q is the left multiplication group,
• Inn(Q) = {f ∈ Mlt(Q) : f (1) = 1} is the inner mapping group,
• Nuc(Q) = {x ∈ Q : x associates with all y, z ∈ Q} is the nucleus,
• Z(Q) = {x ∈ Nuc(Q) : xy = yx for all y ∈ Q} is the center,

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 6 / 66

Basic definitions (since this is the first lecture)

• An algebra (Q, ·, \, /) is a quasigroup if

x · (x\y) = y, x\(x · y) = y, (x · y)/y = x, (x/y) · y = x.

• A quasigroup is a loop if there is 1 ∈ Q such that 1 · x = x · 1 = x.
• Mlt(Q) = Lx, Rx : x ∈ Q is the mutiplication group,
• Mltℓ(Q) = Lx : x ∈ Q is the left multiplication group,
• Inn(Q) = {f ∈ Mlt(Q) : f (1) = 1} is the inner mapping group,
• Nuc(Q) = {x ∈ Q : x associates with all y, z ∈ Q} is the nucleus,
• Z(Q) = {x ∈ Nuc(Q) : xy = yx for all y ∈ Q} is the center,
• A(Q), the associator subloop is the smallest normal subloop such

that Q/A(Q) is a group.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 6 / 66

Outline

1 Introduction

Outline

1 Introduction 2 Quandles

Coloring arcs of oriented knots Knot quandles and the Yang-Baxter equation Asymptotic growth and enumeration results Main ingredients of the enumeration Connected quandles

Outline

1 Introduction 2 Quandles

Coloring arcs of oriented knots Knot quandles and the Yang-Baxter equation Asymptotic growth and enumeration results Main ingredients of the enumeration Connected quandles

3 Bruck loops

Correspondences Bruck loops of odd prime power order The case p3

Outline

1 Introduction 2 Quandles

Coloring arcs of oriented knots Knot quandles and the Yang-Baxter equation Asymptotic growth and enumeration results Main ingredients of the enumeration Connected quandles

3 Bruck loops

Correspondences Bruck loops of odd prime power order The case p3

4 Other recent enumeration results

Bol loops of order pq Small distributive and medial quasigroups

Outline

1 Introduction 2 Quandles

Coloring arcs of oriented knots Knot quandles and the Yang-Baxter equation Asymptotic growth and enumeration results Main ingredients of the enumeration Connected quandles

3 Bruck loops

Correspondences Bruck loops of odd prime power order The case p3

4 Other recent enumeration results

Bol loops of order pq Small distributive and medial quasigroups

Introduction

The objects

We will be enumerating and/or classifying up to isomorphism various algebras with a nonassociative operation.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 9 / 66

Introduction

The objects

We will be enumerating and/or classifying up to isomorphism various algebras with a nonassociative operation.

1 quandles and racks

• provide a complete invariant of oriented knots up to mirror image
• form a class of set-theoretical solutions of the Yang-Baxter equation
• to classify them, it is helpful to first classify subgroups of symmetric

groups up to conjugation

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 9 / 66

Introduction

The objects

We will be enumerating and/or classifying up to isomorphism various algebras with a nonassociative operation.

1 quandles and racks

• provide a complete invariant of oriented knots up to mirror image
• form a class of set-theoretical solutions of the Yang-Baxter equation
• to classify them, it is helpful to first classify subgroups of symmetric

groups up to conjugation

2 Bruck loops

• relativistic addition of 3-vectors results in a Bruck loop
• studied by Glauberman (his Z∗-theorem originated in a paper on Bruck

loops)

• those of order pk resemble p-groups

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 9 / 66

Introduction

The objects

We will be enumerating and/or classifying up to isomorphism various algebras with a nonassociative operation.

1 quandles and racks

• provide a complete invariant of oriented knots up to mirror image
• form a class of set-theoretical solutions of the Yang-Baxter equation
• to classify them, it is helpful to first classify subgroups of symmetric

groups up to conjugation

2 Bruck loops

• relativistic addition of 3-vectors results in a Bruck loop
• studied by Glauberman (his Z∗-theorem originated in a paper on Bruck

loops)

• those of order pk resemble p-groups

3 distributive, medial and trimedial quasigroups

• among the oldest classes of quasigroups considered,
• afford affine representations over abelian groups and commutative

Moufang loops.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 9 / 66

Introduction

The methods

Challenges:

• deficient or non-existent theory of presentations
• no standard representation theory (permutations, matrices)
• explicit isomorphism checks are slow

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 10 / 66

Introduction

The methods

Challenges:

• deficient or non-existent theory of presentations
• no standard representation theory (permutations, matrices)
• explicit isomorphism checks are slow

Main ideas:

• reduce the problem to group actions or, better, linear algebra
• act by a suitable group G on a suitable parameter space X; study
• rbits, stabilizers and invariant subsets:

|X| =

• x∈X/G

|G| |Gx|, |X/G| = 1 |G|

• g∈G

|X g|

• develop extension theory, central extensions, cocycles; solve large

systems of linear equations

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 10 / 66

Outline

1 Introduction 2 Quandles

Coloring arcs of oriented knots Knot quandles and the Yang-Baxter equation Asymptotic growth and enumeration results Main ingredients of the enumeration Connected quandles

3 Bruck loops

Correspondences Bruck loops of odd prime power order The case p3

4 Other recent enumeration results

Bol loops of order pq Small distributive and medial quasigroups

Quandles | Coloring arcs of oriented knots

Coloring rules

Color a diagram of an oriented knot K by an algebra (X, ⊳, ) according to these rules:

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 12 / 66

Quandles | Coloring arcs of oriented knots

Coloring rules

Color a diagram of an oriented knot K by an algebra (X, ⊳, ) according to these rules: x y y ⊳ x y x y x

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 12 / 66

Quandles | Coloring arcs of oriented knots

Coloring rules

Color a diagram of an oriented knot K by an algebra (X, ⊳, ) according to these rules: x y y ⊳ x y x y x Which properties must hold for ⊳, so that the coloring be invariant under Reidemeister moves?

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 12 / 66

Quandles | Coloring arcs of oriented knots

Reidemeister I

There are many oriented Reidemeister moves, but all are combinations of the following five.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 13 / 66

Quandles | Coloring arcs of oriented knots

Reidemeister I

There are many oriented Reidemeister moves, but all are combinations of the following five. x x x ⊳ x x x x x

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 13 / 66

Quandles | Coloring arcs of oriented knots

Reidemeister I

There are many oriented Reidemeister moves, but all are combinations of the following five. x x x ⊳ x x x x x So far we have x ⊳ x = x = x x.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 13 / 66

Quandles | Coloring arcs of oriented knots

Reidemeister II

x y x (y ⊳ x) x y y ⊳ x x y y (y x) ⊳ x x y x

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 14 / 66

Quandles | Coloring arcs of oriented knots

Reidemeister II

x y x (y ⊳ x) x y y ⊳ x x y y (y x) ⊳ x x y x So far we have x ⊳ x = x = x x, (y ⊳ x) x = y and (y x) ⊳ x = y. Hence R⊳

x = (R x )−1 and we don’t need to keep track of anymore.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 14 / 66

Quandles | Coloring arcs of oriented knots

Reidemeister III

y ⊳ x (z ⊳ y) ⊳ x x z ⊳ y y z y ⊳ x (z ⊳ x) ⊳ (y ⊳ x) x y z z ⊳ x

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 15 / 66

Quandles | Coloring arcs of oriented knots

Reidemeister III

y ⊳ x (z ⊳ y) ⊳ x x z ⊳ y y z y ⊳ x (z ⊳ x) ⊳ (y ⊳ x) x y z z ⊳ x Altogether, we have: (X, ⊳) such that x ⊳ x = x and Rx ∈ Aut(X, ⊳).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 15 / 66

Quandles | Knot quandles and the Yang-Baxter equation

Quandles and racks

Definition

A groupoid (Q, ·) is a (right) rack if

• Rx is a bijection of Q for every x ∈ Q,
• (yx)(zx) = (yz)x for every x, y, z ∈ Q.

A rack (Q, ·) is a quandle if

• xx = x for every x ∈ Q.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 16 / 66

Quandles | Knot quandles and the Yang-Baxter equation

Knot quandles

• The quandle freely generated by arcs of K with presenting relations

corresponding to the coloring rules is the knot quandle of Joyce and

• Matveev. It is a complete invariant of oriented knots up to mirror

image.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 17 / 66

Quandles | Knot quandles and the Yang-Baxter equation

Knot quandles

• The quandle freely generated by arcs of K with presenting relations

corresponding to the coloring rules is the knot quandle of Joyce and

• Matveev. It is a complete invariant of oriented knots up to mirror

image.

• Not all assignments of quandle elements to arcs are consistent.

Counting possible colorings by a given finite quandle is a good invariant of oriented knots.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 17 / 66

Quandles | Knot quandles and the Yang-Baxter equation

Knot quandles

• The quandle freely generated by arcs of K with presenting relations

corresponding to the coloring rules is the knot quandle of Joyce and

• Matveev. It is a complete invariant of oriented knots up to mirror

image.

• Not all assignments of quandle elements to arcs are consistent.

Counting possible colorings by a given finite quandle is a good invariant of oriented knots.

• Some non-quandles might give a consistent coloring, e.g., when not

all elements are used as colors.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 17 / 66

Quandles | Knot quandles and the Yang-Baxter equation

Set-theoretical solutions to the Yang-Baxter equation

The Yang-Baxter equation is the equation (σ ⊗ 1)(1 ⊗ σ)(σ ⊗ 1) = (1 ⊗ σ)(σ ⊗ 1)(1 ⊗ σ) (YBE) in any context where the syntax makes sense.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 18 / 66

Quandles | Knot quandles and the Yang-Baxter equation

Set-theoretical solutions to the Yang-Baxter equation

The Yang-Baxter equation is the equation (σ ⊗ 1)(1 ⊗ σ)(σ ⊗ 1) = (1 ⊗ σ)(σ ⊗ 1)(1 ⊗ σ) (YBE) in any context where the syntax makes sense. A set-theoretical solution is any function σ : X × X → X × X such that (YBE) holds as an equality of functions X × X × X → X × X × X.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 18 / 66

Quandles | Knot quandles and the Yang-Baxter equation

Set-theoretical solutions to the Yang-Baxter equation

The Yang-Baxter equation is the equation (σ ⊗ 1)(1 ⊗ σ)(σ ⊗ 1) = (1 ⊗ σ)(σ ⊗ 1)(1 ⊗ σ) (YBE) in any context where the syntax makes sense. A set-theoretical solution is any function σ : X × X → X × X such that (YBE) holds as an equality of functions X × X × X → X × X × X. Given a quandle (X, ⊳), the function σ(x, y) = (y, x ⊳ y) is a set-theoretical solution.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 18 / 66

Quandles | Knot quandles and the Yang-Baxter equation

Set-theoretical solutions to the Yang-Baxter equation

The Yang-Baxter equation is the equation (σ ⊗ 1)(1 ⊗ σ)(σ ⊗ 1) = (1 ⊗ σ)(σ ⊗ 1)(1 ⊗ σ) (YBE) in any context where the syntax makes sense. A set-theoretical solution is any function σ : X × X → X × X such that (YBE) holds as an equality of functions X × X × X → X × X × X. Given a quandle (X, ⊳), the function σ(x, y) = (y, x ⊳ y) is a set-theoretical solution.

• David Stanovsk´

y will report on this and other classes of set-theoretical solutions of the Yang-Baxter equation.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 18 / 66

Quandles | Asymptotic growth and enumeration results

Asymptotic growth

Let q(n) denote the number of quandles of order n up to isomorphism, and r(n) the number of quandles up to isomorphism.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 19 / 66

Quandles | Asymptotic growth and enumeration results

Asymptotic growth

Let q(n) denote the number of quandles of order n up to isomorphism, and r(n) the number of quandles up to isomorphism.

Theorem (Blackburn 2013)

For all sufficiently large orders n, we have 2n2/4−o(n log(n)) ≤ q(n) ≤ r(n) ≤ 2cn2, where c is a constant approximately equal to 1.5566.

Theorem (Ashford and Riordan 2017)

For every ε > 0 and for all sufficiently large orders n we have 2n2/4−ε ≤ q(n) ≤ r(n) ≤ 2n2/4+ε.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 19 / 66

Quandles | Asymptotic growth and enumeration results

Enumeration results

n q(n) r(n) comments 1 1 1 2 1 2 3 3 6 4 7 19 5 22 74 6 73 353 7 298 2080 easy; add column, test, backtrack

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 20 / 66

Quandles | Asymptotic growth and enumeration results

Enumeration results

n q(n) r(n) comments 1 1 1 2 1 2 3 3 6 4 7 19 5 22 74 6 73 353 7 298 2080 easy; add column, test, backtrack 8 1581 16023 McCarron 9 11079 159526 q(9) McCarron

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 20 / 66

Quandles | Asymptotic growth and enumeration results

Enumeration results

n q(n) r(n) comments 1 1 1 2 1 2 3 3 6 4 7 19 5 22 74 6 73 353 7 298 2080 easy; add column, test, backtrack 8 1581 16023 McCarron 9 11079 159526 q(9) McCarron 10 102771 2093244

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 20 / 66

Quandles | Asymptotic growth and enumeration results

Enumeration results

n q(n) r(n) comments 1 1 1 2 1 2 3 3 6 4 7 19 5 22 74 6 73 353 7 298 2080 easy; add column, test, backtrack 8 1581 16023 McCarron 9 11079 159526 q(9) McCarron 10 102771 2093244 11 1275419 36265070

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 20 / 66

Quandles | Asymptotic growth and enumeration results

Enumeration results

n q(n) r(n) comments 1 1 1 2 1 2 3 3 6 4 7 19 5 22 74 6 73 353 7 298 2080 easy; add column, test, backtrack 8 1581 16023 McCarron 9 11079 159526 q(9) McCarron 10 102771 2093244 11 1275419 36265070 12 21101335 836395102

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 20 / 66

Quandles | Asymptotic growth and enumeration results

Enumeration results

n q(n) r(n) comments 1 1 1 2 1 2 3 3 6 4 7 19 5 22 74 6 73 353 7 298 2080 easy; add column, test, backtrack 8 1581 16023 McCarron 9 11079 159526 q(9) McCarron 10 102771 2093244 11 1275419 36265070 12 21101335 836395102 BUT WAIT, THERE IS MORE

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 20 / 66

Quandles | Asymptotic growth and enumeration results

Enumeration results

n q(n) r(n) comments 1 1 1 2 1 2 3 3 6 4 7 19 5 22 74 6 73 353 7 298 2080 easy; add column, test, backtrack 8 1581 16023 McCarron 9 11079 159526 q(9) McCarron 10 102771 2093244 11 1275419 36265070 12 21101335 836395102 BUT WAIT, THERE IS MORE 13 469250886 25794670618 V + Yang

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 20 / 66

Quandles | Main ingredients of the enumeration

Isomorphisms and conjugation I

Let’s switch to left racks and quandles. So Lx are bijections and x(yz) = (xy)(xz) holds.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 21 / 66

Quandles | Main ingredients of the enumeration

Isomorphisms and conjugation I

Let’s switch to left racks and quandles. So Lx are bijections and x(yz) = (xy)(xz) holds.

Definition

For a rack X let Mltℓ(X) = Lx : x ∈ X ≤ SX be the left multiplication group of X.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 21 / 66

Quandles | Main ingredients of the enumeration

Isomorphisms and conjugation II

Proposition (folklore for left quasigroups, explicitly in V+Y)

Let X be a set. (i) If (X, ∗), (X, ◦) are isomorphic racks then Mltℓ(X, ∗), Mltℓ(X, ◦) are conjugate subgroups of SX. (ii) Let G, H be conjugate subgroups of SX. Then the set of racks on X with left multiplication group equal to G contains the same isomorphism types as the set of racks on X with left multiplication group equal to H. (iii) Let (X, ∗), (X, ◦) be two racks with Mltℓ(X, ∗) = G = Mltℓ(X, ◦). Then (X, ∗), (X, ◦) are isomorphic if and only if there is an isomorphism f : (X, ∗) → (X, ◦) satisfying f ∈ NSX (G).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 22 / 66

Quandles | Main ingredients of the enumeration

Conjugacy classes of subgroups of symmetric groups

It is a nontrivial problem to calculate subgroups of Sn up to conjugation. The following takes several hours in GAP:

n 1 2 3 4 5 6 7 8 9 10 11 12 13 s(n) 1 2 4 11 19 56 96 296 554 1593 3094 10723 20832

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 23 / 66

Quandles | Main ingredients of the enumeration

Conjugacy classes of subgroups of symmetric groups

It is a nontrivial problem to calculate subgroups of Sn up to conjugation. The following takes several hours in GAP:

n 1 2 3 4 5 6 7 8 9 10 11 12 13 s(n) 1 2 4 11 19 56 96 296 554 1593 3094 10723 20832

State of the art:

Theorem (Holt)

There are 7598016157515302757 subgroups of S18, partitioned into 7274651 conjugacy classes.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 23 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes I

• Let G be a subgroup of SX.
• Let X/G be orbit representatives of the natural action of G on X.
• A rack on X with G = Mltℓ(X) ≤ Aut(X) is determined by

(Lx : x ∈ X/G):

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 24 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes I

• Let G be a subgroup of SX.
• Let X/G be orbit representatives of the natural action of G on X.
• A rack on X with G = Mltℓ(X) ≤ Aut(X) is determined by

(Lx : x ∈ X/G): Indeed, if y = xg for some g ∈ G then zLy = zLxg = (xg)z = (x · zg−1)g = zLg

x

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 24 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes I

• Let G be a subgroup of SX.
• Let X/G be orbit representatives of the natural action of G on X.
• A rack on X with G = Mltℓ(X) ≤ Aut(X) is determined by

(Lx : x ∈ X/G): Indeed, if y = xg for some g ∈ G then zLy = zLxg = (xg)z = (x · zg−1)g = zLg

x

Which tuples Λ = (λx ∈ SX : x ∈ X/G) correspond to racks on X satisfying λx = Lx for every x ∈ X/G and Mltℓ(X) = G? Call such (G, Λ) a rack envelope.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 24 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes II

Theorem (Blackburn, V+Y)

Let G ≤ SX and Λ = (λx ∈ SX : x ∈ X/G). Then (G, Λ) is a rack envelope iff (i) λx ∈ CG(Gx) for every x ∈ X/G, and (ii)

x∈X/G λG x generates G.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 25 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes II

Theorem (Blackburn, V+Y)

Let G ≤ SX and Λ = (λx ∈ SX : x ∈ X/G). Then (G, Λ) is a rack envelope iff (i) λx ∈ CG(Gx) for every x ∈ X/G, and (ii)

x∈X/G λG x generates G.

Proof.

⇐: We must set Ly = λgy

x , where gy ∈ G is such that xgy = y.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 25 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes II

Theorem (Blackburn, V+Y)

Let G ≤ SX and Λ = (λx ∈ SX : x ∈ X/G). Then (G, Λ) is a rack envelope iff (i) λx ∈ CG(Gx) for every x ∈ X/G, and (ii)

x∈X/G λG x generates G.

Proof.

⇐: We must set Ly = λgy

x , where gy ∈ G is such that xgy = y.

Well-defined: xg = xh implies gh−1 ∈ Gx so λgh−1

x

= λx by (i).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 25 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes II

Theorem (Blackburn, V+Y)

Let G ≤ SX and Λ = (λx ∈ SX : x ∈ X/G). Then (G, Λ) is a rack envelope iff (i) λx ∈ CG(Gx) for every x ∈ X/G, and (ii)

x∈X/G λG x generates G.

Proof.

⇐: We must set Ly = λgy

x , where gy ∈ G is such that xgy = y.

Well-defined: xg = xh implies gh−1 ∈ Gx so λgh−1

x

= λx by (i). Mltℓ(X) = λG

x : x ∈ X/G = G by (ii).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 25 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes II

Theorem (Blackburn, V+Y)

Let G ≤ SX and Λ = (λx ∈ SX : x ∈ X/G). Then (G, Λ) is a rack envelope iff (i) λx ∈ CG(Gx) for every x ∈ X/G, and (ii)

x∈X/G λG x generates G.

Proof.

⇐: We must set Ly = λgy

x , where gy ∈ G is such that xgy = y.

Well-defined: xg = xh implies gh−1 ∈ Gx so λgh−1

x

= λx by (i). Mltℓ(X) = λG

x : x ∈ X/G = G by (ii).

Rack: For u, v, w let x, gv ∈ G be such that xgv = v. Then xgvLu = vLu = u ∗ v so Lu∗v = λgvLu

x

= (λgv

x )Lu = LLu v , so

(u ∗ v) ∗ (u ∗ w) = wLuLu∗v = wLuLLu

v = wLvLu = u ∗ (v ∗ w).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 25 / 66

Quandles | Main ingredients of the enumeration

Rack and quandle envelopes III

Theorem (V+Y, special case by Hulpke + Stanovsk´ y + V)

Let G ≤ SX and Λ = (λx ∈ SX : x ∈ X/G). Then (G, Λ) is a quandle envelope iff (i) λx ∈ Z(Gx) for every x ∈ X/G, and (ii)

x∈X/G λG x generates G.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 26 / 66

Quandles | Main ingredients of the enumeration

Action on parameter spaces

For a group G ≤ SX let Parr(G) =

• x∈X/G

CG(Gx), Parq(G) =

• x∈X/G

Z(Gx).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 27 / 66

Quandles | Main ingredients of the enumeration

Action on parameter spaces

For a group G ≤ SX let Parr(G) =

• x∈X/G

CG(Gx), Parq(G) =

• x∈X/G

Z(Gx). The isomorphism relation induces an action of NSX (G) on Parr(G) as follows: Given f ∈ NSX (G) and (κx : x ∈ X/G) = (λx : x ∈ X/G)f , we have κx = ((λyg−1

y )gy )f

for every x ∈ X/G, where y = xf −1 and zgy = y for every z ∈ X/G.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 27 / 66

Quandles | Main ingredients of the enumeration

Visualizing the action on rack/quandle folders

For f ∈ NSX (G) construct a digraph Γr(G, f ) as follows:

• vertex set is the formal disjoint union of CG(Gx) for x ∈ X/G,
• there is an edge λz → κx iff y = xf −1, z = yg−1

y

and κx = ((λz)gy )f .

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 28 / 66

Quandles | Main ingredients of the enumeration

Visualizing the action on rack/quandle folders

For f ∈ NSX (G) construct a digraph Γr(G, f ) as follows:

• vertex set is the formal disjoint union of CG(Gx) for x ∈ X/G,
• there is an edge λz → κx iff y = xf −1, z = yg−1

y

and κx = ((λz)gy )f . Every vertex has outdegree equal to 1.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 28 / 66

Quandles | Main ingredients of the enumeration

Visualizing the action on rack/quandle folders

For f ∈ NSX (G) construct a digraph Γr(G, f ) as follows:

• vertex set is the formal disjoint union of CG(Gx) for x ∈ X/G,
• there is an edge λz → κx iff y = xf −1, z = yg−1

y

and κx = ((λz)gy )f . Every vertex has outdegree equal to 1. To see the action of f , select one vertex in each CG(Gx) and follow the edges.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 28 / 66

Quandles | Main ingredients of the enumeration

Example

X = {1, . . . , 5}, G = (1, 2)(3, 4, 5) ∼ = C6, f = (1, 2)(4, 5).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 29 / 66

Quandles | Main ingredients of the enumeration

Example

X = {1, . . . , 5}, G = (1, 2)(3, 4, 5) ∼ = C6, f = (1, 2)(4, 5). Then X/G = {1, 3} and we get:

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 29 / 66

Quandles | Main ingredients of the enumeration

Another example

X = {1, . . . , 7}, G = (1, 2), (1, 2, 3), (4, 5), (4, 5, 6) ∼ = S3 × S3, f = (1, 5)(2, 4)(3, 6).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 30 / 66

Quandles | Main ingredients of the enumeration

Another example

X = {1, . . . , 7}, G = (1, 2), (1, 2, 3), (4, 5), (4, 5, 6) ∼ = S3 × S3, f = (1, 5)(2, 4)(3, 6). Then X/G = {1, 4, 7} and we get:

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 30 / 66

Quandles | Main ingredients of the enumeration

Another example

X = {1, . . . , 7}, G = (1, 2), (1, 2, 3), (4, 5), (4, 5, 6) ∼ = S3 × S3, f = (1, 5)(2, 4)(3, 6). Then X/G = {1, 4, 7} and we get:

• Fixpoints of f are easy to see. Namely, a selection of λx ∈ CG(Gx)

that form a cycle in each connected component.

• Hence fixpoints are easily counted and Burnside’s Lemma applies.
• Unfortunately, this does not work for envelopes because

λG

x : x ∈ X/G = G must be tested in each case.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 30 / 66

Quandles | Main ingredients of the enumeration

Comments on the action

Difficulties:

• Parr(G) can be large, especially if G is an elementary abelian

2-group. There is a nonabelian G ≤ S13 for which Parr(G) has over 2 billion elements.

• Not every (λG

x : x ∈ X/G) ∈ Parr(G) generates G. This must be

explicitly tested. Indexing breaks down on the relevant subset.

• Not clear how to use Burnside’s Lemma efficiently for envelopes.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 31 / 66

Quandles | Main ingredients of the enumeration

Comments on the action

Difficulties:

• Parr(G) can be large, especially if G is an elementary abelian

2-group. There is a nonabelian G ≤ S13 for which Parr(G) has over 2 billion elements.

• Not every (λG

x : x ∈ X/G) ∈ Parr(G) generates G. This must be

explicitly tested. Indexing breaks down on the relevant subset.

• Not clear how to use Burnside’s Lemma efficiently for envelopes.

What was done:

• Careful indexing and ad hoc orbit calculations to save memory.
• Calculating the action of f : κx depends only on y = xf −1 and λyg−1

y ,

so the action can be precalculated on “pairs” rather than on |X/G|-tuples.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 31 / 66

Quandles | Main ingredients of the enumeration

Results

The algorithm:

• confirms all previously known results r(≤ 8), q(≤ 9) in 3 seconds,

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 32 / 66

Quandles | Main ingredients of the enumeration

Results

The algorithm:

• confirms all previously known results r(≤ 8), q(≤ 9) in 3 seconds,
• takes about a day to find isomorphism types for r(11) and q(12),

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 32 / 66

Quandles | Main ingredients of the enumeration

Results

The algorithm:

• confirms all previously known results r(≤ 8), q(≤ 9) in 3 seconds,
• takes about a day to find isomorphism types for r(11) and q(12),
• crashes on r(12), r(13) and q(13),

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 32 / 66

Quandles | Main ingredients of the enumeration

Results

The algorithm:

• confirms all previously known results r(≤ 8), q(≤ 9) in 3 seconds,
• takes about a day to find isomorphism types for r(11) and q(12),
• crashes on r(12), r(13) and q(13),
• takes 3 weeks to determine isomorphism types of racks of order 13

with nonabelian left multiplication groups.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 32 / 66

Quandles | Main ingredients of the enumeration

Results

The algorithm:

• confirms all previously known results r(≤ 8), q(≤ 9) in 3 seconds,
• takes about a day to find isomorphism types for r(11) and q(12),
• crashes on r(12), r(13) and q(13),
• takes 3 weeks to determine isomorphism types of racks of order 13

with nonabelian left multiplication groups.

Lemma

A rack X is 2-reductive (that is, (xy)y = y) if and only if Mltℓ(X) is abelian. Jedliˇ cka, Pilitowska, Stanovsk´ y and Zamojska-Dzienio used affine meshes to construct all 2-reductive racks, in principle. They use Burnside’s Lemma efficiently to count 2-reductive racks up to n ≤ 14. Using their counts for the abelian case, we determined r(12), r(13) and q(13).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 32 / 66

Quandles | Connected quandles

Connected quandles

A quandle X is connected iff Mltℓ(X) acts transitively on X.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 33 / 66

Quandles | Connected quandles

Connected quandles

A quandle X is connected iff Mltℓ(X) acts transitively on X.

Theorem (Hulpke, Stanovsk´ y, V)

There is a one-to-one correspondence between connected quandles with Mltℓ(X) = G and quandle envelopes (G, (λx)), where λx ∈ Z(Gx) and λG

x = G.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 33 / 66

Quandles | Connected quandles

Connected quandles

A quandle X is connected iff Mltℓ(X) acts transitively on X.

Theorem (Hulpke, Stanovsk´ y, V)

There is a one-to-one correspondence between connected quandles with Mltℓ(X) = G and quandle envelopes (G, (λx)), where λx ∈ Z(Gx) and λG

x = G.

Search for connected quandles was carried out independently by H+S+V and by Leandro Vendramin (University of Buenos Aires).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 33 / 66

Quandles | Connected quandles

Connected quandles: Results

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 q(n) 1 1 1 3 2 5 3 8 1 9 10 11 7 9 ℓ(n) 1 1 1 3 5 2 8 9 1 11 5 9 a(n) 1 1 1 3 5 2 8 9 1 11 3 9 n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 q(n) 15 12 17 10 9 21 42 34 65 13 27 24 29 17 ℓ(n) 15 17 3 7 21 2 34 62 7 27 29 8 a(n) 15 17 3 5 21 2 34 30 5 27 29 8 n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 q(n) 11 15 73 35 13 33 39 26 41 9 45 45 ℓ(n) 11 15 9 35 13 6 39 41 9 36 45 a(n) 9 15 8 35 11 6 39 41 9 24 45

Table: The numbers q(n) of connected quandles, ℓ(n) of latin quandles, and a(n)

• f connected affine quandles of size n ≤ 47 up to isomorphism.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 34 / 66

Outline

1 Introduction 2 Quandles

Coloring arcs of oriented knots Knot quandles and the Yang-Baxter equation Asymptotic growth and enumeration results Main ingredients of the enumeration Connected quandles

3 Bruck loops

Correspondences Bruck loops of odd prime power order The case p3

4 Other recent enumeration results

Bol loops of order pq Small distributive and medial quasigroups

Bruck loops Correspondences

Latin and involutory quandles

Definition

A quandle (Q, ·) is

• latin if also all right translations Rx : Q → Q, y → yx are bijections
• f Q,

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 36 / 66

Bruck loops Correspondences

Latin and involutory quandles

Definition

A quandle (Q, ·) is

• latin if also all right translations Rx : Q → Q, y → yx are bijections
• f Q,
• involutory if L2

x = 1, i.e., x(xy) = y for every x, y ∈ Q.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 36 / 66

Bruck loops Correspondences

Division notation

In a latin quandle, we will denote by x\y = L−1

x (y)

the left division operation and by y/x = R−1

x (y)

the right division operation.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 37 / 66

Bruck loops Correspondences

Division notation

In a latin quandle, we will denote by x\y = L−1

x (y)

the left division operation and by y/x = R−1

x (y)

the right division operation.

• In an involutory quandle, we have L−1

x

= Lx due to L2

x = 1, and thus

x\y = xy.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 37 / 66

Bruck loops Correspondences

Division notation

In a latin quandle, we will denote by x\y = L−1

x (y)

the left division operation and by y/x = R−1

x (y)

the right division operation.

• In an involutory quandle, we have L−1

x

= Lx due to L2

x = 1, and thus

x\y = xy.

• A finite involutory quandle is necessarily of odd order. (Proof:

Consider orbits of any given Lx.)

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 37 / 66

Bruck loops Correspondences

Bol and Bruck loops

Definition

A loop (Q, ·) is (left) Bol if x(y(xz)) = (x(yx))z. Equivalently, LxLyLx is a left translation for every x, y ∈ Q.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 38 / 66

Bruck loops Correspondences

Bol and Bruck loops

Definition

A loop (Q, ·) is (left) Bol if x(y(xz)) = (x(yx))z. Equivalently, LxLyLx is a left translation for every x, y ∈ Q.

Definition

A loop (Q, ·) is (left) Bruck if it is left Bol and satisfies (xy)−1 = x−1y−1.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 38 / 66

Bruck loops Correspondences

Notes on Bruck loops

• Bruck loops form a well-studied variety of loops.
• They motivated Glauberman to prove several key results for the

classification of finite simple groups.

• Three-dimensional vectors under Einstein relativistic vector addition

form a non-associative Bruck loop.

• Due to the below correspondence with involutory latin quandles,

uniquely 2-divisible Bruck loops can also be seen as solutions to (YBE).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 39 / 66

Bruck loops Correspondences

Notes on Bruck loops

• Bruck loops form a well-studied variety of loops.
• They motivated Glauberman to prove several key results for the

classification of finite simple groups.

• Three-dimensional vectors under Einstein relativistic vector addition

form a non-associative Bruck loop.

• Due to the below correspondence with involutory latin quandles,

uniquely 2-divisible Bruck loops can also be seen as solutions to (YBE). A groupoid (Q, ·) is uniquely 2-divisible if the squaring map x → x2 is a bijection of Q.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 39 / 66

Bruck loops Correspondences

Involutory latin quandles versus U2D Bruck loops

Theorem (Kikkawa, Robinson)

Let Q be a set and let e ∈ Q. There is a one-to-one correspondence between involutory latin quandles defined on Q and uniquely 2-divisible Bruck loops defined on Q with identity element e. In more detail: (i) If (Q, ·) is an involutory latin quandle then (Q, +) defined by x + y = (x/e)(e\y) = (x/e)(ey) is a uniquely 2-divisible Bruck loop with identity element e. (ii) If (Q, +) is a uniquely 2-divisible Bruck loop with identity e then (Q, ·) defined by xy = (x + x) − y = 2x − y is an involutory latin quandle. (iii) The two mappings are mutual inverses.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 40 / 66

Bruck loops Correspondences

Automorphic loops and Γ-loops

Definition

A loop Q is automorphic of Inn(Q) ≤ Aut(Q). Lots of recent results on automorphic loops by Grishkov, Jedliˇ cka, Kinyon, Nagy, V.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 41 / 66

Bruck loops Correspondences

Automorphic loops and Γ-loops

Definition

A loop Q is automorphic of Inn(Q) ≤ Aut(Q). Lots of recent results on automorphic loops by Grishkov, Jedliˇ cka, Kinyon, Nagy, V.

Definition

Let Px = L−1

x−1Rx. A Γ-loop is a commutative loop with automorphic

inverse property (xy)−1 = x−1y−1 satisfying LxLx−1 = Lx−1Lx and PxPyPx = PPx(y).

Theorem (Greer)

Every Γ-loop is power associative.

• see the talk of Lee Raney

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 41 / 66

Bruck loops Correspondences

Bruck loops of odd order versus Γ-loops of odd order

Theorem (Greer)

There is a one-to-one correspondence between left Bruck loops of odd order n and Γ-loops of odd order n. In more detail: (i) If (Q, +) is a left Bruck loop of odd order n with identity element e then (Q, ·) defined by x · y = (LxLyL−1

x L−1 y )1/2LyLx(e)

is a Γ-loop of order n. Here, Lx(y) = x + y. (ii) If (Q, ·) is a Γ-loop of odd order n then (Q, +) defined by x + y = (x−1\(y 2x))1/2 is a left Bruck loop of order n. (iii) The mappings of (i) and (ii) are mutual inverses.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 42 / 66

Bruck loops Correspondences

Remarks on the correspondence

• Γ-loops of odd order n contain all commutative automorphic loops of
• rder n,
• Kinyon and Greer constructed a Bruck loop whose associated Γ-loop

is not automorphic.

Problem

For which orders n, particulary odd prime power orders n = pk, are all Γ-loops automorphic?

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 43 / 66

Bruck loops Bruck loops of odd prime power order

Central nilpotence for loops

Definition

The center Z(Q) of a loop Q consists of all elements that commute and associative with all elements of Q.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 44 / 66

Bruck loops Bruck loops of odd prime power order

Central nilpotence for loops

Definition

The center Z(Q) of a loop Q consists of all elements that commute and associative with all elements of Q. A loop Q is centrally nilpotent if the sequence Q0 = Q, Q1 = Q0/Z(Q0), Q2 = Q1/Z(Q1), ... reaches the trivial loop in finitely many steps.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 44 / 66

Bruck loops Bruck loops of odd prime power order

Central nilpotence for loops

Definition

The center Z(Q) of a loop Q consists of all elements that commute and associative with all elements of Q. A loop Q is centrally nilpotent if the sequence Q0 = Q, Q1 = Q0/Z(Q0), Q2 = Q1/Z(Q1), ... reaches the trivial loop in finitely many steps.

Theorem (Glauberman)

Bruck loops of odd prime power order pk are centrally nilpotent.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 44 / 66

Bruck loops Bruck loops of odd prime power order

Central extensions

Definition

A loop Q is a central extension of an abelian group A by a loop F if A ≤ Z(Q) and Q/A ∼ = F.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 45 / 66

Bruck loops Bruck loops of odd prime power order

Central extensions

Definition

A loop Q is a central extension of an abelian group A by a loop F if A ≤ Z(Q) and Q/A ∼ = F.

Theorem (Bruck)

A loop Q is a central extension of an abelian group A by a loop F if and

• nly if Q ∼

= (F × A, ∗), where (x, a) ∗ (y, b) = (xy, a + b + θ(x, y)) for some θ : F × F → A satisfying θ(1, x) = θ(x, 1) = 0 for all x ∈ F.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 45 / 66

Bruck loops Bruck loops of odd prime power order

Bruck cocycles

We can construct all centrally nilpotent loops of order pk as central extensions of A = Zp by a centrally nilpotent loop of order pk−1.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 46 / 66

Bruck loops Bruck loops of odd prime power order

Bruck cocycles

We can construct all centrally nilpotent loops of order pk as central extensions of A = Zp by a centrally nilpotent loop of order pk−1.

Proposition

Let A be an abelian group and F a loop. Then (F × A, ∗) is a Bruck loop if and only if F is a Bruck loop and θ : F × F → A satisfies θ(x, z) + θ(y, xz) + θ(x, y(xz)) = θ(y, x) + θ(x, yx) + θ(x(yx), z), θ(x, x−1) + θ(y, y−1) = θ(x, y) + θ(x−1, y−1) + θ(xy, (xy)−1).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 46 / 66

Bruck loops Bruck loops of odd prime power order

Coboundaries

• Denote by C(F, A) all Bruck cocycles.
• Let B(F, A) be the coboundaries, that is, cocycles

θ(x, y) = τ(xy) − τ(x) − τ(y) for some τ : F → A satisfying τ(1) = 0.

• Let H(F, A) = C(F, A)/B(F, A).
• If θ − µ ∈ B(F, A) then the corresponding loops are isomorphic (but

not vice versa).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 47 / 66

Bruck loops Bruck loops of odd prime power order

Action of automorphism groups on cohomology

The group Aut(F) × Aut(A) acts on H(F, A) via θ(α,β)(x, y) = β−1(θ(α(x), α(y))).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 48 / 66

Bruck loops Bruck loops of odd prime power order

Action of automorphism groups on cohomology

The group Aut(F) × Aut(A) acts on H(F, A) via θ(α,β)(x, y) = β−1(θ(α(x), α(y))). The loops corresponding to θ and θ(α,β) are isomorphic.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 48 / 66

Bruck loops Bruck loops of odd prime power order

Action of automorphism groups on cohomology

The group Aut(F) × Aut(A) acts on H(F, A) via θ(α,β)(x, y) = β−1(θ(α(x), α(y))). The loops corresponding to θ and θ(α,β) are isomorphic. Up to isomorphism, it therefore suffices to consider representatives of

• rbits of Aut(F) × Aut(A) on H(F, A).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 48 / 66

Bruck loops Bruck loops of odd prime power order

Computational issues

• With A = Zp, the cocycle condition can be restated as a system of

linear equations in unknowns θ(x, y).

• There are O(|F|3) equations in |F|2 unknowns. The system is

solvable with |F| ≤ 100 or so.

• The orbits of the action of Aut(F) × Aut(A) are computed by

converting everything to a permutation action; this can be slow.

• Constructed loops must be filtered up to isomorphism, a very difficult
• task. We used the isomorphism filters in the LOOPS package for GAP.
• If |Z(Q)| > p, Q can be obtained from two or more different factor

loops.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 49 / 66

Bruck loops Bruck loops of odd prime power order

Results for p = 3

Theorem (Stuhl and V 2017)

The number of Bruck loops of order 33, 34, 35 up to isomorphism is, respectively, 7, 72 , ≥ 118763.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 50 / 66

Bruck loops Bruck loops of odd prime power order

Results for p = 3

Theorem (Stuhl and V 2017)

The number of Bruck loops of order 33, 34, 35 up to isomorphism is, respectively, 7, 72 , ≥ 118763.

• The only case we were not able to do are the central extensions of Z3

by the elementary abelian group Z4

3.

• We know that dim H(Z4

3, Z3) = 24. This is presently not tractable.

• It took several months of computing time to get to this point.
• We found Bruck loops of order 35 whose associated Γ-loop is not

automorphic.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 50 / 66

Bruck loops The case p3

Overview of results for small exponents

Let p be an odd prime.

Theorem (Burn)

Bruck loops of order p2 are groups.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 51 / 66

Bruck loops The case p3

Overview of results for small exponents

Let p be an odd prime.

Theorem (Burn)

Bruck loops of order p2 are groups.

Theorem (de Barros, Grishkov and V)

There are 7 commutative automorphic loops of order p3 and hence at least 7 Bruck loops of order p3 up to isomorphism.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 51 / 66

Bruck loops The case p3

Overview of results for small exponents

Let p be an odd prime.

Theorem (Burn)

Bruck loops of order p2 are groups.

Theorem (de Barros, Grishkov and V)

There are 7 commutative automorphic loops of order p3 and hence at least 7 Bruck loops of order p3 up to isomorphism.

Proof.

• All such loops are centrally nilpotent.
• All are homomorphic images of the free 2-generator comm. automorphic

loop F of class 2, in fact of a certain factor Fp of F.

• The group GL2(p) acts on Fp, and isomorphic loops correspond to orbits of

this action. Count the orbits!

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 51 / 66

Bruck loops The case p3

Overview of results for small exponents

Theorem (Bianco + Bonato 2019)

There are precisely 7 Bruck loops of odd order p3 up to isomorphism, p

• dd. Consequently, all Γ-loops of odd order p3 are Γ-loops.

Proof.

Main idea: Understand the structure of the displacement group LaL−1

b

: a, b ∈ X for involutory quandles of order p3.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 52 / 66

Bruck loops The case p3

Associator calculus in commutative automorphic loops of nilpotency class 2

Lemma

Let Q be a commutative loop of nilpotency class two. Then: (i) A(Q) ≤ Z(Q). (ii) (a, b, a) = 1, (a, b, c) = (c, b, a)−1, and (a, b, c)(b, c, a)(c, a, b) = 1 for every a, b, c ∈ Q. (iii) Q is an automorphic loop if and only if (ab, c, d) = (a, c, d)(b, c, d) for every a, b, c, d ∈ Q.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 53 / 66

Bruck loops The case p3

Associator calculus in commutative automorphic loops of nilpotency class 2

Lemma

Let Q be a commutative automorphic loop of nilpotency class two. Then: (i) For every a, b, c, d ∈ Q, (ab, c, d) = (a, c, d)(b, c, d), (a, b, cd) = (a, b, c)(a, b, d), (a, bc, d) = (a, d, b)(a, d, c)(b, a, d)(c, a, d). (ii) For every a, b, c, d ∈ Q, (ab)(cd) = (ac)(bd)(ac, b, d)(b, a, c)(d, c, ab).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 54 / 66

Bruck loops The case p3

Associator calculus II

Lemma

(iii) For every a, b ∈ Q and i, j, k ∈ Z, (ai, bj, bk) = (a, b, b)ijk, (bi, aj, bk) = 1, (bi, bj, ak) = (b, b, a)ijk. (iv) For every a, b ∈ Q and i1, i2, j1, j2, k1, k2 ∈ Z, (ai1bi2, aj1bj2, ak1bk2) = (a, a, b)j1(i1k2−i2k1)(a, b, b)j2(i1k2−i2k1).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 55 / 66

Bruck loops The case p3

The free 2-generator comm. automorphic loop of class 2

Theorem

Let F be the free commutative automorphic loop of nilpotency class two with free generators x1, x2, and let z1 = (x1, x1, x2), z2 = (x1, x2, x2). Then every element of F can be written uniquely as xa1

1 xa2 2 za3 1 za4 2

for some a1, a2, a3, a4 ∈ Z, and the multiplication in F is given by (xa1

1 xa2 2 za3 1 za4 2 )(xb1 1 xb2 2 zb3 1 zb4 2 ) = xa1+b1 1

xa2+b2

2

za3+b3−a1b1(a2+b2)

1

za4+b4+a2b2(a1+b1)

2

. Furthermore, Z(F) = Nλ(F) = Nµ(F) = N(F) = A(F) = z1, z2 ∼ = Z2. The loop F is the central extension of the free abelian group z1, z2 by the free abelian group with free generators x1, x2 via the cocycle θ(xa1

1 xa2 2 , xb1 1 xb2 2 ) = z−a1b1(a2+b2) 1

za2b2(a1+b1)

2

. Finally, the associator in F is given by (xa1

1 xa2 2 za3 1 za4 2 , xb1 1 xb2 2 zb3 1 zb4 2 , xc1 1 xc2 2 zc3 1 zc4 2 ) = zb1(a1c2−a2c1) 1

zb2(a1c2−a2c1)

2

.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 56 / 66

Bruck loops The case p3

The free 2-generator comm. automorphic loop of class 2

Theorem

Let F be the free commutative automorphic loop of nilpotency class two with free generators x1, x2, and let z1 = (x1, x1, x2), z2 = (x1, x2, x2). Then every element of F can be written uniquely as xa1

1 xa2 2 za3 1 za4 2

for some a1, a2, a3, a4 ∈ Z, and the multiplication in F is given by (xa1

1 xa2 2 za3 1 za4 2 )(xb1 1 xb2 2 zb3 1 zb4 2 ) = xa1+b1 1

xa2+b2

2

za3+b3−a1b1(a2+b2)

1

za4+b4+a2b2(a1+b1)

2

. Furthermore, Z(F) = Nλ(F) = Nµ(F) = N(F) = A(F) = z1, z2 ∼ = Z2. The loop F is the central extension of the free abelian group z1, z2 by the free abelian group with free generators x1, x2 via the cocycle θ(xa1

1 xa2 2 , xb1 1 xb2 2 ) = z−a1b1(a2+b2) 1

za2b2(a1+b1)

2

. Finally, the associator in F is given by (xa1

1 xa2 2 za3 1 za4 2 , xb1 1 xb2 2 zb3 1 zb4 2 , xc1 1 xc2 2 zc3 1 zc4 2 ) = zb1(a1c2−a2c1) 1

zb2(a1c2−a2c1)

2

.

• Dylene Agda Souza de Barros will report on the free 2-generator comm.

automorphic loop of class 3.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 56 / 66

Outline

1 Introduction 2 Quandles

Coloring arcs of oriented knots Knot quandles and the Yang-Baxter equation Asymptotic growth and enumeration results Main ingredients of the enumeration Connected quandles

3 Bruck loops

Correspondences Bruck loops of odd prime power order The case p3

4 Other recent enumeration results

Bol loops of order pq Small distributive and medial quasigroups

Other recent enumeration results Bol loops of order pq

Bol loops of order pq up to isomorphism

At the 2017 MileHigh conference, Kinyon and Nagy reported this result with a complicated proof:

Theorem (Kinyon, Nagy, V)

Let p > q be odd primes. (i) A nonassociative right Bol loop of order pq exists if and only if q divides p2 − 1. (ii) If q divides p2 − 1, there exists a unique nonassociative right Bruck loop Bp,q of order pq up to isomorphism, and there are precisely (p − q + 4)/2 right Bol loops of order pq up to isomorphism.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 58 / 66

Other recent enumeration results Bol loops of order pq

Bol loops of order pq up to isomorphism

Theorem (K+N+V)

(iii) If q divides p2 − 1, the (p − q + 4)/2 right Bol loops of order pq can be constructed on Fq × Fp with multiplication (i, j)(k, ℓ) = (i + k, ℓ(1 + θk)−1 + (j + ℓ(1 + θk)−1)θ−1

i

θi+k), where θ0, . . . , θq−1 ∈ Fp are chosen as follows. Fix a non-square t of Fp, write Fp2 = {u + v√t : u, v ∈ Fp}, and let ω ∈ Fp2 be a primitive qth root of unity. Let Γ = {γ ∈ Fp : 1 ≤ γ ≤ (p + 1)/2, 1 − γ−1 ∈ ω}, q|p − 1, {γ = 1/2 + m√t : 0 ≤ m ≤ (p − 1)/2, 1 − γ−1 ∈ ω}, q|p + 1, be a set of cardinality (p − q + 2)/2. Then either let θi = 1 for every i ∈ Fq,

• r choose γ ∈ Γ and let θi = (γωi + (1 − γ)ω−i)−1 ∈ Fp for every i ∈ Fq.

The choice γ = 1/2 = (p + 1)/2 results in the nonassociative right Bruck loop Bp,q.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 59 / 66

Other recent enumeration results Bol loops of order pq

Bol loops of order pq up to isomorphism

Theorem (K+N+V)

(iv) Let Q be a nonassociative right Bol loop of order pq. Then Q contains a unique subloop of order p and this subloop is normal and equal to the left nucleus of Q. The right nucleus and the middle nucleus of Q are trivial. The right multiplication group of Q has order p2q or p3q. (v) The right multiplication group of Bp,q is isomorphic to (Zp × Zp) ⋊ Zq.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 60 / 66

Other recent enumeration results Bol loops of order pq

Bol loops of order pq up to isotopism

Theorem (V 2018)

Let p > q be odd primes such that q divides p2 − 1. Then there are precisely p − 1 + 4q 2q

• Bol loops of order pq up to isotopism. With the notation of the previous

theorem, these loops are obtained as follows: Set θi = 1 for every i ∈ Fq for the cyclic group of order pq. The non-cyclic loops correspond to orbit representatives of the group f , g acting on Γp,q, where f (γ) = 1 − γ and g(γ) = γω γω + (1 − γ)ω−1 .

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 61 / 66

Other recent enumeration results Small distributive and medial quasigroups

Some classes of quasigroups

A quasigroup is:

• medial or entropic if (xy)(uv) = (xu)(yv) holds,
• trimedial if any three of its elements generate a medial quasigroup,
• distributive if x(yz) = (xy)(xz) and (xy)z = (xz)(yz) hold,
• Moufang if x(y(xz)) = ((xy)x)z holds.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 62 / 66

Other recent enumeration results Small distributive and medial quasigroups

Some classes of quasigroups

A quasigroup is:

• medial or entropic if (xy)(uv) = (xu)(yv) holds,
• trimedial if any three of its elements generate a medial quasigroup,
• distributive if x(yz) = (xy)(xz) and (xy)z = (xz)(yz) hold,
• Moufang if x(y(xz)) = ((xy)x)z holds.

Theorem (Belousov)

A quasigroup is distributive iff it is trimedial and idempotent.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 62 / 66

Other recent enumeration results Small distributive and medial quasigroups

Affine quasigroups

Let (G, +) be a loop, φ, ψ ∈ Aut(G) and c ∈ Z(G). Define (G, ∗) by x ∗ y = φ(x) + ψ(y) + c. Then the quasigroup (G, ∗) = Q(G, +, φ, ψ, c) is said to be affine over the loop (G, +).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 63 / 66

Other recent enumeration results Small distributive and medial quasigroups

Affine representations

Theorem (Toyoda-Murdoch-Bruck)

Medial quasigroups are precisely the quasigroups affine over abelian groups with φψ = ψφ.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 64 / 66

Other recent enumeration results Small distributive and medial quasigroups

Affine representations

Theorem (Toyoda-Murdoch-Bruck)

Medial quasigroups are precisely the quasigroups affine over abelian groups with φψ = ψφ.

Theorem (Kepka)

Trimedial quasigroups are precisely the quasigroups affine over commutative Moufang loops with φψ = ψφ such that both φ, ψ are

• central. (An automorphism α of (G, +) is central if

Z(G) + x = Z(G) + α(x) for every x ∈ G.)

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 64 / 66

Other recent enumeration results Small distributive and medial quasigroups

Affine representations

Theorem (Toyoda-Murdoch-Bruck)

Medial quasigroups are precisely the quasigroups affine over abelian groups with φψ = ψφ.

Theorem (Kepka)

Trimedial quasigroups are precisely the quasigroups affine over commutative Moufang loops with φψ = ψφ such that both φ, ψ are

• central. (An automorphism α of (G, +) is central if

Z(G) + x = Z(G) + α(x) for every x ∈ G.)

Theorem (Belousov-Soublin)

Distributive quasigroups are precisely the quasigroups affine over commutative Moufang loops with φ, ψ central and φ = id − ψ (so φψ = ψφ for free).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 64 / 66

Other recent enumeration results Small distributive and medial quasigroups

The isomorphism problem in affine representations

There are theorems due to Dr´ apal and Kepka that effectively solve the isomorphism problems for medial, paramedial and distributive quasigroups in their affine representations. For instance:

Theorem (Dr´ apal)

Medial quasigroups Q(G, +, φ1, ψ1, c1), Q(G, +, φ2, ψ2, c2) are isomorphic iff there are γ ∈ Aut(G, +) and u ∈ Im(id − φ1 − ψ1) such that φ1 = φγ

2,

ψ1 = ψγ

2 and c2 = γ(c1 + u).

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Other recent enumeration results Small distributive and medial quasigroups

Enumeration of affine quasigroups: Results

Theorem (Stanovsk´ y + V)

Enumeration of medial quasigroups of order < 128 except for those isotopic to C4 × C 4

2 , C 6 2 , C 4 3 and C 3 5 . (For instance, there are 4193 medial

quasigroups of order 32.)

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 66 / 66

Other recent enumeration results Small distributive and medial quasigroups

Enumeration of affine quasigroups: Results

Theorem (Stanovsk´ y + V)

Enumeration of medial quasigroups of order < 128 except for those isotopic to C4 × C 4

2 , C 6 2 , C 4 3 and C 3 5 . (For instance, there are 4193 medial

quasigroups of order 32.)

Theorem (Jedliˇ cka, Stanovsk´ y + V)

There are 17004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, B´ en´ etau and Lacaze), 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Nˇ emec), and 6 non-medial distributive Mendelsohn quasigroups of order 243 (extending the work of Donovan, Griggs, McCourt, Oprˇ sal and Stanovsky´ y).

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 66 / 66

Other recent enumeration results Small distributive and medial quasigroups

Enumeration of affine quasigroups: Results

Theorem (Stanovsk´ y + V)

Enumeration of medial quasigroups of order < 128 except for those isotopic to C4 × C 4

2 , C 6 2 , C 4 3 and C 3 5 . (For instance, there are 4193 medial

quasigroups of order 32.)

Theorem (Jedliˇ cka, Stanovsk´ y + V)

There are 17004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, B´ en´ etau and Lacaze), 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Nˇ emec), and 6 non-medial distributive Mendelsohn quasigroups of order 243 (extending the work of Donovan, Griggs, McCourt, Oprˇ sal and Stanovsky´ y).

• ˇ

Zaneta Semaniˇ sinov´ a will report on enumeration of paramedial quasigroups (affine with φ2 = ψ2) of order p and p2.

Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 66 / 66