Fun with F Vaughan Jones, Vanderbilt, Berkeley, Auckland August 9, - - PowerPoint PPT Presentation

Fun with F

Vaughan Jones, Vanderbilt, Berkeley, Auckland August 9, 2015

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 1 / 10

Motivation.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 2 / 10

index supertransitivity 4 5 3+ √ 5 6 6

1 5

×∞ D(1)

n+2

• ne ∞-depth

E(1)

6

E(1)

7

E(1)

8

×2 ×2 ×4 at least one ∞-depth ×1 ×3 ×∞ unclassifiably many ∞-depth

∞

A∞ at every index

Hyperfinite A∞ at the index of E10 ×2 E6 ×2 E8

A series D s e r i e s

1 2 (5 +

√ 13)

1 2(5 +

√ 17) 3 + √ 3

1 2 (5 +

√ 21) ×3 ×3

Peters, Morrison, Snyder, Izumi, Bigelow, Penneys, Tener, Calegari,Liu, Grossman,...

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 3 / 10

index supertransitivity 4 5 3+ √ 5 6 6

1 5

×∞ D(1)

n+2

• ne ∞-depth

E(1)

6

E(1)

7

E(1)

8

×2 ×2 ×4 at least one ∞-depth ×1 ×3 ×∞ unclassifiably many ∞-depth

∞

A∞ at every index

Hyperfinite A∞ at the index of E10 ×2 E6 ×2 E8

A series D s e r i e s

1 2 (5 +

√ 13)

1 2(5 +

√ 17) 3 + √ 3

1 2 (5 +

√ 21) ×3 ×3

Peters, Morrison, Snyder, Izumi, Bigelow, Penneys, Tener, Calegari,Liu, Grossman,... The big question: Do they all ”come from” conformal field theory?

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 3 / 10

Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff (S1).

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff (S1). Locally (Haag-Kastler) the family of Diff (I) where I is an interval on S1.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff (S1). Locally (Haag-Kastler) the family of Diff (I) where I is an interval on S1.

Definition

The Thompson group T is the group of all orientation preserving homeomorphisms of S1 which are piecewise linear (lifted to R), smooth except at finitely many dyadic rationals ({ m

2n : m, n ∈ Z}), and whose

slopes, when they exist, are powers of 2.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff (S1). Locally (Haag-Kastler) the family of Diff (I) where I is an interval on S1.

Definition

The Thompson group T is the group of all orientation preserving homeomorphisms of S1 which are piecewise linear (lifted to R), smooth except at finitely many dyadic rationals ({ m

2n : m, n ∈ Z}), and whose

slopes, when they exist, are powers of 2. The Thompson group F is the subgroup of T preserving 0 and 1.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff (S1). Locally (Haag-Kastler) the family of Diff (I) where I is an interval on S1.

Definition

The Thompson group T is the group of all orientation preserving homeomorphisms of S1 which are piecewise linear (lifted to R), smooth except at finitely many dyadic rationals ({ m

2n : m, n ∈ Z}), and whose

slopes, when they exist, are powers of 2. The Thompson group F is the subgroup of T preserving 0 and 1. Provided the end points of I are dyadic rationals, the set of elements of T that are the identity outside I is isomorphic to F. As a first shot at answering the big question, we construct ”toy” models where Diff (S1), Diff (I) is replaced by T, F.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff (S1). Locally (Haag-Kastler) the family of Diff (I) where I is an interval on S1.

Definition

The Thompson group T is the group of all orientation preserving homeomorphisms of S1 which are piecewise linear (lifted to R), smooth except at finitely many dyadic rationals ({ m

2n : m, n ∈ Z}), and whose

slopes, when they exist, are powers of 2. The Thompson group F is the subgroup of T preserving 0 and 1. Provided the end points of I are dyadic rationals, the set of elements of T that are the identity outside I is isomorphic to F. As a first shot at answering the big question, we construct ”toy” models where Diff (S1), Diff (I) is replaced by T, F. Ghys,Sergiescu.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

Ingredient of CFT is a representation of the Virasoro algebra, or a projective representation of Diff (S1). Locally (Haag-Kastler) the family of Diff (I) where I is an interval on S1.

Definition

The Thompson group T is the group of all orientation preserving homeomorphisms of S1 which are piecewise linear (lifted to R), smooth except at finitely many dyadic rationals ({ m

2n : m, n ∈ Z}), and whose

slopes, when they exist, are powers of 2. The Thompson group F is the subgroup of T preserving 0 and 1. Provided the end points of I are dyadic rationals, the set of elements of T that are the identity outside I is isomorphic to F. As a first shot at answering the big question, we construct ”toy” models where Diff (S1), Diff (I) is replaced by T, F. Ghys,Sergiescu.F conjugate to a subgroup of Diff ([0, 1]) ( in Homeo(S1)).

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 4 / 10

For this talk we will just construct ”unitary” representations of F in a knot theoretic context (Conway tangles) which define honest unitary representations using some data of a subfactor.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 5 / 10

For this talk we will just construct ”unitary” representations of F in a knot theoretic context (Conway tangles) which define honest unitary representations using some data of a subfactor. The empty tangle will be the vacuum vector Ω and the vacuum expectation value gΩ, Ω for g ∈ F will be an unoriented link in R3.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 5 / 10

For this talk we will just construct ”unitary” representations of F in a knot theoretic context (Conway tangles) which define honest unitary representations using some data of a subfactor. The empty tangle will be the vacuum vector Ω and the vacuum expectation value gΩ, Ω for g ∈ F will be an unoriented link in R3. In fact by the linearization device ”group algebra, GNS” we need only show how to associate a link to an element of F. Honest unitarity of course requires a lot more.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 5 / 10

For this talk we will just construct ”unitary” representations of F in a knot theoretic context (Conway tangles) which define honest unitary representations using some data of a subfactor. The empty tangle will be the vacuum vector Ω and the vacuum expectation value gΩ, Ω for g ∈ F will be an unoriented link in R3. In fact by the linearization device ”group algebra, GNS” we need only show how to associate a link to an element of F. Honest unitarity of course requires a lot more. So the fun starts.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 5 / 10

Bifurcating trees.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 6 / 10

Bifurcating trees. Elements of F are given by pairs of (planar rooted) bifurcating trees T with the same number of leaves.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 6 / 10

Bifurcating trees. Elements of F are given by pairs of (planar rooted) bifurcating trees T with the same number of leaves. This is because each such tree T specifies a partition of [0, 1] into dyadic intervals thus:

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 6 / 10

Bifurcating trees. Elements of F are given by pairs of (planar rooted) bifurcating trees T with the same number of leaves. This is because each such tree T specifies a partition of [0, 1] into dyadic intervals thus:

Intervals of I (0,1/2)

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 6 / 10

So given a pair T1, T2 as above we can get an element of F simply by mapping the intervals specified by the leaves of T1 in the unique affine way onto those of T2.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 7 / 10

So given a pair T1, T2 as above we can get an element of F simply by mapping the intervals specified by the leaves of T1 in the unique affine way onto those of T2. We represent the Thompson group element by a diagram with T1 on top, T2 beneath.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 7 / 10

So given a pair T1, T2 as above we can get an element of F simply by mapping the intervals specified by the leaves of T1 in the unique affine way onto those of T2. We represent the Thompson group element by a diagram with T1 on top, T2 beneath.The simplest nontrivial example is:

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 7 / 10

The link gΩ, Ω is obtained simply from the pair of trees by the ”caret gambit”:

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 8 / 10

example: → →

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 9 / 10

example: → → A lot of simplification always occurs in the resulting link diagram.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 9 / 10

example: → → A lot of simplification always occurs in the resulting link diagram.In the above example one obtains the simplest non-trivial 2-component link often called the Hopf link.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 9 / 10

”Alexander-type” theorem:

Theorem

Any (unoriented) knot or link can be obtained as gΩ, Ω.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 10 / 10

”Alexander-type” theorem:

Theorem

Any (unoriented) knot or link can be obtained as gΩ, Ω. For the oriented case one may define a subgroup F of F as the set of all g for which the surface (defined by the caret gambit) whose boundary is the link gΩ, Ω is orientable.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 10 / 10

”Alexander-type” theorem:

Theorem

Any (unoriented) knot or link can be obtained as gΩ, Ω. For the oriented case one may define a subgroup F of F as the set of all g for which the surface (defined by the caret gambit) whose boundary is the link gΩ, Ω is orientable. An improvement on the above theorem says that any oriented link may be obtained as gΩ, Ω for g ∈ F.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 10 / 10

”Alexander-type” theorem:

Theorem

Any (unoriented) knot or link can be obtained as gΩ, Ω. For the oriented case one may define a subgroup F of F as the set of all g for which the surface (defined by the caret gambit) whose boundary is the link gΩ, Ω is orientable. An improvement on the above theorem says that any oriented link may be obtained as gΩ, Ω for g ∈ F. Thus F is as good as the braid groups at producing knots and links!

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 10 / 10

”Alexander-type” theorem:

Theorem

Any (unoriented) knot or link can be obtained as gΩ, Ω. For the oriented case one may define a subgroup F of F as the set of all g for which the surface (defined by the caret gambit) whose boundary is the link gΩ, Ω is orientable. An improvement on the above theorem says that any oriented link may be obtained as gΩ, Ω for g ∈ F. Thus F is as good as the braid groups at producing knots and links! This reinforces the conviction that F and B∞ have a lot in common-e.g. Mc Duff, Haagerup property.

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 10 / 10

”Alexander-type” theorem:

Theorem

Any (unoriented) knot or link can be obtained as gΩ, Ω. For the oriented case one may define a subgroup F of F as the set of all g for which the surface (defined by the caret gambit) whose boundary is the link gΩ, Ω is orientable. An improvement on the above theorem says that any oriented link may be obtained as gΩ, Ω for g ∈ F. Thus F is as good as the braid groups at producing knots and links! This reinforces the conviction that F and B∞ have a lot in common-e.g. Mc Duff, Haagerup property. Surprising result of Golan and Sapir: F ∼ = F3 (Thompson 3, not free group.)

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 10 / 10

”Alexander-type” theorem:

Theorem

Any (unoriented) knot or link can be obtained as gΩ, Ω. For the oriented case one may define a subgroup F of F as the set of all g for which the surface (defined by the caret gambit) whose boundary is the link gΩ, Ω is orientable. An improvement on the above theorem says that any oriented link may be obtained as gΩ, Ω for g ∈ F. Thus F is as good as the braid groups at producing knots and links! This reinforces the conviction that F and B∞ have a lot in common-e.g. Mc Duff, Haagerup property. Surprising result of Golan and Sapir: F ∼ = F3 (Thompson 3, not free group.) Missing is a ”Markov” type theorem-when do two elements of F (or F) produce the same link?

Vaughan Jones, Vanderbilt, Berkeley, Auckland Fun with F August 9, 2015 10 / 10