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 orbits, stabilizers and invariant subsets: | G | 1 � � | X g | | X | = | G x | , | X / G | = | G | x ∈ X / G g ∈ 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 p 3 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: y ⊳ x y � x y y x 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: y ⊳ x y � x y y x 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 y x y x ( y � x ) ⊳ x y ⊳ x y � x y x ( y ⊳ x ) � x y x Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 14 / 66
Quandles | Coloring arcs of oriented knots Reidemeister II y x y x ( y � x ) ⊳ x y ⊳ x y � x y x ( y ⊳ x ) � x y x So far we have x ⊳ x = x = x � x , ( y ⊳ x ) � x = y and ( y � x ) ⊳ x = y . x ) − 1 and we don’t need to keep track of � anymore. Hence R ⊳ x = ( R � Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 14 / 66
Quandles | Coloring arcs of oriented knots Reidemeister III y y z z x z ⊳ x z ⊳ y x y ⊳ x ( z ⊳ y ) ⊳ x y ⊳ x ( z ⊳ x ) ⊳ ( y ⊳ x ) Petr Vojtˇ echovsk´ y (University of Denver) Enumeration Loops ’19 15 / 66
Quandles | Coloring arcs of oriented knots Reidemeister III y y z z x z ⊳ x z ⊳ y x y ⊳ x ( z ⊳ y ) ⊳ x y ⊳ x ( z ⊳ x ) ⊳ ( y ⊳ x ) Altogether, we have: ( X , ⊳ ) such that x ⊳ x = x and R x ∈ 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 • R x 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 2 n 2 / 4 − o ( n log( n )) ≤ q ( n ) ≤ r ( n ) ≤ 2 cn 2 , 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 2 n 2 / 4 − ε ≤ q ( n ) ≤ r ( n ) ≤ 2 n 2 / 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 L x 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 L x are bijections and x ( yz ) = ( xy )( xz ) holds. Definition For a rack X let Mlt ℓ ( X ) = � L x : x ∈ X � ≤ S X 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 Mlt ℓ ( X , ∗ ) , Mlt ℓ ( X , ◦ ) are conjugate subgroups of S X . (ii) Let G, H be conjugate subgroups of S X . 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 Mlt ℓ ( X , ∗ ) = G = Mlt ℓ ( X , ◦ ) . Then ( X , ∗ ) , ( X , ◦ ) are isomorphic if and only if there is an isomorphism f : ( X , ∗ ) → ( X , ◦ ) satisfying f ∈ N S X ( 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 S n up to conjugation. The following takes several hours in GAP: 1 2 3 4 5 6 7 8 9 10 11 12 13 n 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 S n up to conjugation. The following takes several hours in GAP: 1 2 3 4 5 6 7 8 9 10 11 12 13 n 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 S 18 , 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 S X . • Let X / G be orbit representatives of the natural action of G on X . • A rack on X with G = Mlt ℓ ( X ) ≤ Aut ( X ) is determined by ( L x : 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 S X . • Let X / G be orbit representatives of the natural action of G on X . • A rack on X with G = Mlt ℓ ( X ) ≤ Aut ( X ) is determined by ( L x : x ∈ X / G ): Indeed, if y = xg for some g ∈ G then zL y = zL xg = ( xg ) z = ( x · zg − 1 ) g = zL g 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 S X . • Let X / G be orbit representatives of the natural action of G on X . • A rack on X with G = Mlt ℓ ( X ) ≤ Aut ( X ) is determined by ( L x : x ∈ X / G ): Indeed, if y = xg for some g ∈ G then zL y = zL xg = ( xg ) z = ( x · zg − 1 ) g = zL g x Which tuples Λ = ( λ x ∈ S X : x ∈ X / G ) correspond to racks on X satisfying λ x = L x for every x ∈ X / G and Mlt ℓ ( 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 ≤ S X and Λ = ( λ x ∈ S X : x ∈ X / G ) . Then ( G , Λ) is a rack envelope iff (i) λ x ∈ C G ( G x ) for every x ∈ X / G, and x ∈ X / G λ G (ii) � 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 ≤ S X and Λ = ( λ x ∈ S X : x ∈ X / G ) . Then ( G , Λ) is a rack envelope iff (i) λ x ∈ C G ( G x ) for every x ∈ X / G, and x ∈ X / G λ G (ii) � x generates G. Proof. ⇐ : We must set L y = λ g y x , where g y ∈ G is such that xg y = 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 ≤ S X and Λ = ( λ x ∈ S X : x ∈ X / G ) . Then ( G , Λ) is a rack envelope iff (i) λ x ∈ C G ( G x ) for every x ∈ X / G, and x ∈ X / G λ G (ii) � x generates G. Proof. ⇐ : We must set L y = λ g y x , where g y ∈ G is such that xg y = y . Well-defined: xg = xh implies gh − 1 ∈ G x so λ gh − 1 = λ x by (i). x 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 ≤ S X and Λ = ( λ x ∈ S X : x ∈ X / G ) . Then ( G , Λ) is a rack envelope iff (i) λ x ∈ C G ( G x ) for every x ∈ X / G, and x ∈ X / G λ G (ii) � x generates G. Proof. ⇐ : We must set L y = λ g y x , where g y ∈ G is such that xg y = y . Well-defined: xg = xh implies gh − 1 ∈ G x so λ gh − 1 = λ x by (i). x Mlt ℓ ( X ) = � λ G 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 ≤ S X and Λ = ( λ x ∈ S X : x ∈ X / G ) . Then ( G , Λ) is a rack envelope iff (i) λ x ∈ C G ( G x ) for every x ∈ X / G, and x ∈ X / G λ G (ii) � x generates G. Proof. ⇐ : We must set L y = λ g y x , where g y ∈ G is such that xg y = y . Well-defined: xg = xh implies gh − 1 ∈ G x so λ gh − 1 = λ x by (i). x Mlt ℓ ( X ) = � λ G x : x ∈ X / G � = G by (ii). Rack: For u , v , w let x , g v ∈ G be such that xg v = v . Then xg v L u = vL u = u ∗ v so L u ∗ v = λ g v L u = ( λ g v x ) L u = L L u v , so x ( u ∗ v ) ∗ ( u ∗ w ) = wL u L u ∗ v = wL u L L u v = wL v L u = 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 ≤ S X and Λ = ( λ x ∈ S X : x ∈ X / G ) . Then ( G , Λ) is a quandle envelope iff (i) λ x ∈ Z ( G x ) for every x ∈ X / G, and x ∈ X / G λ G (ii) � 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 ≤ S X let � � Par r ( G ) = C G ( G x ) , Par q ( G ) = Z ( G x ) . x ∈ X / G x ∈ X / G 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 ≤ S X let � � Par r ( G ) = C G ( G x ) , Par q ( G ) = Z ( G x ) . x ∈ X / G x ∈ X / G The isomorphism relation induces an action of N S X ( G ) on Par r ( G ) as follows: Given f ∈ N S X ( G ) and ( κ x : x ∈ X / G ) = ( λ x : x ∈ X / G ) f , we have y ) g y ) f κ x = (( λ yg − 1 for every x ∈ X / G , where y = xf − 1 and zg y = 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 ∈ N S X ( G ) construct a digraph Γ r ( G , f ) as follows: • vertex set is the formal disjoint union of C G ( G x ) for x ∈ X / G , and κ x = (( λ z ) g y ) f . • there is an edge λ z → κ x iff y = xf − 1 , z = yg − 1 y 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 ∈ N S X ( G ) construct a digraph Γ r ( G , f ) as follows: • vertex set is the formal disjoint union of C G ( G x ) for x ∈ X / G , and κ x = (( λ z ) g y ) f . • there is an edge λ z → κ x iff y = xf − 1 , z = yg − 1 y 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 ∈ N S X ( G ) construct a digraph Γ r ( G , f ) as follows: • vertex set is the formal disjoint union of C G ( G x ) for x ∈ X / G , and κ x = (( λ z ) g y ) f . • there is an edge λ z → κ x iff y = xf − 1 , z = yg − 1 y Every vertex has outdegree equal to 1. To see the action of f , select one vertex in each C G ( G x ) 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) � ∼ = C 6 , 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) � ∼ = C 6 , 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) � ∼ = S 3 × S 3 , 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) � ∼ = S 3 × S 3 , 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) � ∼ = S 3 × S 3 , 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 λ x ∈ C G ( G x ) 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 � λ G 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 : • Par r ( G ) can be large, especially if G is an elementary abelian 2-group. There is a nonabelian G ≤ S 13 for which Par r ( G ) has over 2 billion elements. • Not every ( λ G x : x ∈ X / G ) ∈ Par r ( 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 : • Par r ( G ) can be large, especially if G is an elementary abelian 2-group. There is a nonabelian G ≤ S 13 for which Par r ( G ) has over 2 billion elements. • Not every ( λ G x : x ∈ X / G ) ∈ Par r ( 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 : κ x depends only on y = xf − 1 and λ yg − 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 Mlt ℓ ( 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 Mlt ℓ ( 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 Mlt ℓ ( X ) acts transitively on X . Theorem (Hulpke, Stanovsk´ y, V) There is a one-to-one correspondence between connected quandles with Mlt ℓ ( X ) = G and quandle envelopes ( G , ( λ x )) , where λ x ∈ Z ( G x ) and � λ G 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 Mlt ℓ ( X ) acts transitively on X . Theorem (Hulpke, Stanovsk´ y, V) There is a one-to-one correspondence between connected quandles with Mlt ℓ ( X ) = G and quandle envelopes ( G , ( λ x )) , where λ x ∈ Z ( G x ) and � λ G 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 0 1 1 3 2 5 3 8 1 9 10 11 0 7 9 ℓ ( n ) 1 0 1 1 3 0 5 2 8 0 9 1 11 0 5 9 a ( n ) 1 0 1 1 3 0 5 2 8 0 9 1 11 0 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 0 21 42 34 0 65 13 27 24 29 17 ℓ ( n ) 15 0 17 3 7 0 21 2 34 0 62 7 27 0 29 8 a ( n ) 15 0 17 3 5 0 21 2 34 0 30 5 27 0 29 8 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 n q ( n ) 11 0 15 73 35 0 13 33 39 26 41 9 45 0 45 ℓ ( n ) 11 0 15 9 35 0 13 6 39 0 41 9 36 0 45 a ( n ) 9 0 15 8 35 0 11 6 39 0 41 9 24 0 45 Table: The numbers q ( n ) of connected quandles, ℓ ( n ) of latin quandles, and a ( n ) of 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 p 3 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 R x : Q → Q , y �→ yx are bijections of 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 R x : Q → Q , y �→ yx are bijections of Q , • involutory if L 2 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 = L x due to L 2 x = 1, and thus x 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 = L x due to L 2 x = 1, and thus x x \ y = xy . • A finite involutory quandle is necessarily of odd order. (Proof: Consider orbits of any given L x .) 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, L x L y L x 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, L x L y L x 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 − 1 y − 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 �→ x 2 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 = 2 x − 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
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