On a Polynomial Alexander Invariant of Tangles and its - - PowerPoint PPT Presentation

On a Polynomial Alexander Invariant of Tangles

and its categorification Claudius Zibrowius

DPMMS, University of Cambridge Supervisor: Dr Jacob Rasmussen

copyECSTATIC 2015copy

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 1 / 10

Table of Contents

• 0. Basics and motivation
• 1. Definition and properties of ∇s

T

• 2. A tangle Floer homology

HFT References

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 1 / 10

Table of Contents

• 0. Basics and motivation
• 1. Definition and properties of ∇s

T

• 2. A tangle Floer homology

HFT References

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 1 / 10

Basics: What are tangles?

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 2 / 10

Basics: What are tangles?

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 2 / 10

Basics: What are tangles?

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 2 / 10

Basics: What are tangles?

b b b b

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 2 / 10

The Alexander polynomial and knot Floer homology

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 3 / 10

The Alexander polynomial and knot Floer homology

∇K(t) ∈ Z[t±1], invariant of oriented knots and links;

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 3 / 10

The Alexander polynomial and knot Floer homology

∇K(t) ∈ Z[t±1], invariant of oriented knots and links;

• e. g. for K =

, ∇K(t) = t2 − 1 + t−2

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 3 / 10

The Alexander polynomial and knot Floer homology

∇K(t) ∈ Z[t±1], invariant of oriented knots and links;

• e. g. for K =

, ∇K(t) = t2 − 1 + t−2 Connected sum formula: ∇K1#K2(t) = ∇K1(t) · ∇K2(t)

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 3 / 10

The Alexander polynomial and knot Floer homology

∇K(t) ∈ Z[t±1], invariant of oriented knots and links;

• e. g. for K =

, ∇K(t) = t2 − 1 + t−2 Connected sum formula: ∇K1#K2(t) = ∇K1(t) · ∇K2(t)

• HFK(K) a doubly graded f. g. Abelian group,

invariant of oriented knots and links;

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 3 / 10

The Alexander polynomial and knot Floer homology

∇K(t) ∈ Z[t±1], invariant of oriented knots and links;

• e. g. for K =

, ∇K(t) = t2 − 1 + t−2 Connected sum formula: ∇K1#K2(t) = ∇K1(t) · ∇K2(t)

• HFK(K) a doubly graded f. g. Abelian group,

invariant of oriented knots and links; Connected sum formula: CFK(K1#K2) = CFK(K1) ⊗ CFK(K2)

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 3 / 10

The Alexander polynomial and knot Floer homology

∇K(t) ∈ Z[t±1], invariant of oriented knots and links;

• e. g. for K =

, ∇K(t) = t2 − 1 + t−2 Connected sum formula: ∇K1#K2(t) = ∇K1(t) · ∇K2(t)

• HFK(K) a doubly graded f. g. Abelian group,

invariant of oriented knots and links; Connected sum formula: CFK(K1#K2) = CFK(K1) ⊗ CFK(K2) χ( HFK(K)) := (−1)hrk( HFK h,a)ta = ∇K(t)

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 3 / 10

Motivation

knot Floer homology Alexander polynomial c a t e g

• r

i fi e s

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Motivation

Khovanov homology knot Floer homology Alexander polynomial Jones polynomial c a t e g

• r

i fi e s c a t e g

• r

i fi e s

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Motivation

Khovanov homology knot Floer homology Alexander polynomial Jones polynomial homology for tangles Bar-Natan’s Khovanov for tangles Jones polynomial categorifies c a t e g

• r

i fi e s c a t e g

• r

i fi e s

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Motivation

Khovanov homology knot Floer homology Alexander polynomial Jones polynomial homology for tangles Bar-Natan’s Khovanov ? for tangles Jones polynomial categorifies c a t e g

• r

i fi e s c a t e g

• r

i fi e s

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Motivation

Khovanov homology knot Floer homology Alexander polynomial Jones polynomial homology for tangles Bar-Natan’s Khovanov ? ? for tangles Jones polynomial categorifies c a t e g

• r

i fi e s c a t e g

• r

i fi e s

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Table of Contents

• 0. Basics and motivation
• 1. Definition and properties of ∇s

T

• 2. A tangle Floer homology

HFT References

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Definition of ∇K for knots and links

b b

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

Definition of ∇K for knots and links

b b b b b

a Kauffman state

Definition

A Kauffman state of a knot diagram K is an assignment of a marker to one of the four regions at each crossing such that each closed region is

• ccupied by exactly one

marker.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

Definition of ∇K for knots and links

b b b b b

a Kauffman state

Definition

A Kauffman state of a knot diagram K is an assignment of a marker to one of the four regions at each crossing such that each closed region is

• ccupied by exactly one

marker.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

Definition of ∇K for knots and links

b b b b b

t 1 t

a labelled Kauffman state

Definition

A Kauffman state of a knot diagram K is an assignment of a marker to one of the four regions at each crossing such that each closed region is

• ccupied by exactly one

marker.

t 1 −t−1 1

Alexander Code for a positive crossing

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

Definition of ∇K for knots and links

b b b b b

t 1 t

∇K(t) = t2 +

Definition

A Kauffman state of a knot diagram K is an assignment of a marker to one of the four regions at each crossing such that each closed region is

• ccupied by exactly one

marker.

t 1 −t−1 1

Alexander Code for a positive crossing

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

Definition of ∇K for knots and links

b b b b b

−t−1 1 t

∇K(t) = t2 − 1 +

Definition

A Kauffman state of a knot diagram K is an assignment of a marker to one of the four regions at each crossing such that each closed region is

• ccupied by exactly one

marker.

t 1 −t−1 1

Alexander Code for a positive crossing

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

Definition of ∇K for knots and links

b b b b b

−t−1 1 −t−1

∇K(t) = t2 − 1 + t−2

Definition

A Kauffman state of a knot diagram K is an assignment of a marker to one of the four regions at each crossing such that each closed region is

• ccupied by exactly one

marker.

t 1 −t−1 1

Alexander Code for a positive crossing

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

Definition of ∇K for knots and links

b b

∇K(t) = t2 − 1 + t−2

Definition

A Kauffman state of a knot diagram K is an assignment of a marker to one of the four regions at each crossing such that each closed region is

• ccupied by exactly one

marker.

t 1 −t−1 1

Alexander Code for a positive crossing

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

Definition of ∇s

T for tangles

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

Definition

A generalised Kauffman state of a tangle diagram T is an assignment of a marker to

• ne of the four regions at each

crossing such that each closed region is occupied by exactly

• ne marker,

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

Definition

A generalised Kauffman state of a tangle diagram T is an assignment of a marker to

• ne of the four regions at each

crossing such that each closed region is occupied by exactly

• ne marker, with the

additional condition that there be at most one marker in each

• pen region.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

t−1 −t 1 1 −t−1

Definition

A generalised Kauffman state of a tangle diagram T is an assignment of a marker to

• ne of the four regions at each

crossing such that each closed region is occupied by exactly

• ne marker, with the

additional condition that there be at most one marker in each

• pen region.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

t−1 −t 1 1 −t−1

Definition

A site s of a 2n-ended tangle is an (n + 1)-element subsets of the set of

• pen regions.

Definition

A generalised Kauffman state of a tangle diagram T is an assignment of a marker to

• ne of the four regions at each

crossing such that each closed region is occupied by exactly

• ne marker, with the

additional condition that there be at most one marker in each

• pen region.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

t−1 −t 1 1 −t−1

Definition

A site s of a 2n-ended tangle is an (n + 1)-element subsets of the set of

• pen regions.

Definition

A generalised Kauffman state of a tangle diagram T is an assignment of a marker to

• ne of the four regions at each

crossing such that each closed region is occupied by exactly

• ne marker, with the

additional condition that there be at most one marker in each

• pen region.

Note: Every Kauffman state “belongs” to a site.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

t−1 −t 1 1 −t−1

Definition

A site s of a 2n-ended tangle is an (n + 1)-element subsets of the set of

• pen regions.

Definition

For each site s of a tangle diagram T, define ∇s

T(t) =

• x∈K(T,s)

c(x), where

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

t−1 −t 1 1 −t−1

Definition

A site s of a 2n-ended tangle is an (n + 1)-element subsets of the set of

• pen regions.

Definition

For each site s of a tangle diagram T, define ∇s

T(t) =

• x∈K(T,s)

c(x), where c(x) is the product of all labellings of the markers of a Kauffman state x and

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Definition of ∇s

T for tangles

b b b b b b b b b

t−1 −t 1 1 −t−1

Definition

A site s of a 2n-ended tangle is an (n + 1)-element subsets of the set of

• pen regions.

Definition

For each site s of a tangle diagram T, define ∇s

T(t) =

• x∈K(T,s)

c(x), where c(x) is the product of all labellings of the markers of a Kauffman state x and K(T, s) is the set of all generalised Kauffman states which have no markers in s.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

Properties of ∇s

T

Theorem (CZ’14)

∇s

T is a tangle invariant.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 7 / 10

Properties of ∇s

T

Theorem (CZ’14)

∇s

T is a tangle invariant.

∇s

T agrees with the Alexander polynomial if T is a knot/link.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 7 / 10

Properties of ∇s

T

Theorem (CZ’14)

∇s

T is a tangle invariant.

∇s

T agrees with the Alexander polynomial if T is a knot/link.

∇s

T can be refined to a multivariate invariant, generalising the

multivariate Alexander polynomial.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 7 / 10

Properties of ∇s

T

Theorem (CZ’14)

∇s

T is a tangle invariant.

∇s

T agrees with the Alexander polynomial if T is a knot/link.

∇s

T can be refined to a multivariate invariant, generalising the

multivariate Alexander polynomial. ∇s

T satisfies a glueing formula which generalises the connected sum

formula for knots and links.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 7 / 10

Properties of ∇s

T

Theorem (CZ’14)

∇s

T is a tangle invariant.

∇s

T agrees with the Alexander polynomial if T is a knot/link.

∇s

T can be refined to a multivariate invariant, generalising the

multivariate Alexander polynomial. ∇s

T satisfies a glueing formula which generalises the connected sum

formula for knots and links. ∇s

T satisfies many properties similar to those of the Alexander

polynomial (see [Har83]). In particular, it is well-behaved under changing orientations and taking mirror images.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 7 / 10

Properties of ∇s

T

Theorem (CZ’14)

∇s

T is a tangle invariant.

∇s

T agrees with the Alexander polynomial if T is a knot/link.

∇s

T can be refined to a multivariate invariant, generalising the

multivariate Alexander polynomial. ∇s

T satisfies a glueing formula which generalises the connected sum

formula for knots and links. ∇s

T satisfies many properties similar to those of the Alexander

polynomial (see [Har83]). In particular, it is well-behaved under changing orientations and taking mirror images. ∇s

T is a mutation invariant.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 7 / 10

Properties of ∇s

T

Theorem (CZ’14)

∇s

T is a tangle invariant.

∇s

T agrees with the Alexander polynomial if T is a knot/link.

∇s

T can be refined to a multivariate invariant, generalising the

multivariate Alexander polynomial. ∇s

T satisfies a glueing formula which generalises the connected sum

formula for knots and links. ∇s

T satisfies many properties similar to those of the Alexander

polynomial (see [Har83]). In particular, it is well-behaved under changing orientations and taking mirror images. ∇s

T is a mutation invariant.

Cor: The multivariate Alexander polynomial is a mutation invariant.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 7 / 10

Table of Contents

• 0. Basics and motivation
• 1. Definition and properties of ∇s

T

• 2. A tangle Floer homology

HFT References

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 7 / 10

Definition of knot Floer homology

Let K ⊂ S3 be a knot and (Σ, α, β, z, w) a Heegaard diagram for K:

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 8 / 10

Definition of knot Floer homology

Let K ⊂ S3 be a knot and (Σ, α, β, z, w) a Heegaard diagram for K: Σ a closed surface

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 8 / 10

Definition of knot Floer homology

Let K ⊂ S3 be a knot and (Σ, α, β, z, w) a Heegaard diagram for K: Σ a closed surface α, β two sets of simple closed curves on Σ

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 8 / 10

Definition of knot Floer homology

Let K ⊂ S3 be a knot and (Σ, α, β, z, w) a Heegaard diagram for K: Σ a closed surface α, β two sets of simple closed curves on Σ z, w two basepoints

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 8 / 10

Definition of knot Floer homology

Let K ⊂ S3 be a knot and (Σ, α, β, z, w) a Heegaard diagram for K: Σ a closed surface α, β two sets of simple closed curves on Σ z, w two basepoints From this, define a chain complex CFK(K):

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 8 / 10

Definition of knot Floer homology

Let K ⊂ S3 be a knot and (Σ, α, β, z, w) a Heegaard diagram for K: Σ a closed surface α, β two sets of simple closed curves on Σ z, w two basepoints From this, define a chain complex CFK(K): Generators intersections of α- and β-curves

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 8 / 10

Definition of knot Floer homology

Let K ⊂ S3 be a knot and (Σ, α, β, z, w) a Heegaard diagram for K: Σ a closed surface α, β two sets of simple closed curves on Σ z, w two basepoints From this, define a chain complex CFK(K): Generators intersections of α- and β-curves Differentials holomorphic curve count

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 8 / 10

Examples: Heegaard diagrams for tangles

b b b b b b b b b b b b b b

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 9 / 10

Examples: Heegaard diagrams for tangles

b b b b

αa β Z∂ D C B A

b b b b b b b b b b

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 9 / 10

Examples: Heegaard diagrams for tangles

b b b b

αa β Z∂ D C B A

b b b b b b b b b b

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 9 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T. Then, we can define a graded chain complex CFT(HT, s), whose graded chain homotopy type is an invariant

• f the tangle T.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T. Then, we can define a graded chain complex CFT(HT, s), whose graded chain homotopy type is an invariant

• f the tangle T. Its graded Euler characteristic is equal to ∇s

T.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T. Then, we can define a graded chain complex CFT(HT, s), whose graded chain homotopy type is an invariant

• f the tangle T. Its graded Euler characteristic is equal to ∇s

T.

Furthermore, if T represents a knot K, HFT(T, s) coincides with the knot Floer homology HFK(K) of K.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T. Then, we can define a graded chain complex CFT(HT, s), whose graded chain homotopy type is an invariant

• f the tangle T. Its graded Euler characteristic is equal to ∇s

T.

Furthermore, if T represents a knot K, HFT(T, s) coincides with the knot Floer homology HFK(K) of K. Note: The differential only counts holomorphic disks away from punctures.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T. Then, we can define a graded chain complex CFT(HT, s), whose graded chain homotopy type is an invariant

• f the tangle T. Its graded Euler characteristic is equal to ∇s

T.

Furthermore, if T represents a knot K, HFT(T, s) coincides with the knot Floer homology HFK(K) of K. Note: The differential only counts holomorphic disks away from punctures. The proof of invariance and d2 = 0 uses sutured Floer homology.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T. Then, we can define a graded chain complex CFT(HT, s), whose graded chain homotopy type is an invariant

• f the tangle T. Its graded Euler characteristic is equal to ∇s

T.

Furthermore, if T represents a knot K, HFT(T, s) coincides with the knot Floer homology HFK(K) of K. Note: The differential only counts holomorphic disks away from punctures. The proof of invariance and d2 = 0 uses sutured Floer homology. No glueing formula for CFT(T, s) can work.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T. Then, we can define a graded chain complex CFT(HT, s), whose graded chain homotopy type is an invariant

• f the tangle T. Its graded Euler characteristic is equal to ∇s

T.

Furthermore, if T represents a knot K, HFT(T, s) coincides with the knot Floer homology HFK(K) of K. Note: The differential only counts holomorphic disks away from punctures. The proof of invariance and d2 = 0 uses sutured Floer homology. No glueing formula for CFT(T, s) can work. Need bordered theory (compare with [PV14]?). . .

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

A categorification of ∇s

T

Theorem (CZ’14)

Let T be a tangle without any closed components, HT a Heegaard diagram for T and s a site of T. Then, we can define a graded chain complex CFT(HT, s), whose graded chain homotopy type is an invariant

• f the tangle T. Its graded Euler characteristic is equal to ∇s

T.

Furthermore, if T represents a knot K, HFT(T, s) coincides with the knot Floer homology HFK(K) of K. Note: The differential only counts holomorphic disks away from punctures. The proof of invariance and d2 = 0 uses sutured Floer homology. No glueing formula for CFT(T, s) can work. Need bordered theory (compare with [PV14]?). . . . . . or some alternative approach (work in progress ).

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

b b b b

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

Table of Contents

• 0. Basics and motivation
• 1. Definition and properties of ∇s

T

• 2. A tangle Floer homology

HFT References

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

Existing approaches

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

Existing approaches

[Arc10] J. Archibald, The multivariable Alexander polynomial on tangles, PhD thesis, University of Toronto, 2010

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

Existing approaches

[Arc10] J. Archibald, The multivariable Alexander polynomial on tangles, PhD thesis, University of Toronto, 2010 [Pol10] M. Polyak, Alexander-Conway invariants of tangles, arXiv: 1011.6200v1

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

Existing approaches

[Arc10] J. Archibald, The multivariable Alexander polynomial on tangles, PhD thesis, University of Toronto, 2010 [Pol10] M. Polyak, Alexander-Conway invariants of tangles, arXiv: 1011.6200v1 [Big12] S. Bigelow, A diagrammatic Alexander invariant of tangles, arXiv: 1203.5457v1 [Ken12] G. Kennedy, A diagrammatic multivariate Alexander invariant

• f tangles, arXiv: 12055781v2

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

Existing approaches

[Arc10] J. Archibald, The multivariable Alexander polynomial on tangles, PhD thesis, University of Toronto, 2010 [Pol10] M. Polyak, Alexander-Conway invariants of tangles, arXiv: 1011.6200v1 [Big12] S. Bigelow, A diagrammatic Alexander invariant of tangles, arXiv: 1203.5457v1 [Ken12] G. Kennedy, A diagrammatic multivariate Alexander invariant

• f tangles, arXiv: 12055781v2

[Kau83] L. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

Existing approaches

[Arc10] J. Archibald, The multivariable Alexander polynomial on tangles, PhD thesis, University of Toronto, 2010 [Pol10] M. Polyak, Alexander-Conway invariants of tangles, arXiv: 1011.6200v1 [Big12] S. Bigelow, A diagrammatic Alexander invariant of tangles, arXiv: 1203.5457v1 [Ken12] G. Kennedy, A diagrammatic multivariate Alexander invariant

• f tangles, arXiv: 12055781v2

[Kau83] L. Kauffman, Formal Knot Theory, Princeton University Press, 1983. [PV14] I. Petkova, V. V´ ertesi, Combinatorial tangle Floer homology arXiv: 1410.2161v1

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10

• J. Archibald, The multivariable Alexander polynomial on tangles, PhD thesis,

University of Toronto, 2010, available at http://www.math.toronto.edu/jfa/jana_thesis.pdf

• S. Bigelow, A diagrammatic Alexander invariant of tangles, arXiv: 1203.5457v1
• D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, arXiv: 0410495v2
• R. Hartley, The Conway potential function for links, Comment. Math. Helvetici, no.

58, pp. 365-378, 1983

• A. Juh´

asz, Holomorphic discs and sutured manifolds, arXiv: 0601443v3

• L. Kauffman, Formal Knot Theory, Princeton University Press, 1983.
• G. Kennedy, A diagrammatic multivariate Alexander invariant of tangles,

arXiv: 12055781v2

• P. Ozsv´

ath, Z. Szab´

• , Holomorphic disks and knot invariants, arXiv: 0209056v4
• P. Ozsv´

ath, Z. Szab´

• , Holomorphic disks and link invariants, arXiv: 0512286v2
• M. Polyak, Alexander-Conway invariants of tangles, arXiv: 1011.6200v1
• I. Petkova, V. V´

ertesi, Combinatorial tangle Floer homology, arXiv: 1410.2161v1

• J. A. Rasmussen, Floer homology and knot complements, arXiv: 0306378v1

Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 10 / 10