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

b b b b Basics: What are tangles? 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; , ∇ K ( t ) = t 2 − 1 + t − 2 e. g. for K = 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; , ∇ K ( t ) = t 2 − 1 + t − 2 e. g. for K = Connected sum formula: ∇ K 1 # K 2 ( t ) = ∇ K 1 ( t ) · ∇ K 2 ( 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; , ∇ K ( t ) = t 2 − 1 + t − 2 e. g. for K = Connected sum formula: ∇ K 1 # K 2 ( t ) = ∇ K 1 ( t ) · ∇ K 2 ( 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; , ∇ K ( t ) = t 2 − 1 + t − 2 e. g. for K = Connected sum formula: ∇ K 1 # K 2 ( t ) = ∇ K 1 ( t ) · ∇ K 2 ( t ) � HFK ( K ) a doubly graded f. g. Abelian group, invariant of oriented knots and links; Connected sum formula: � CFK ( K 1 # K 2 ) = � CFK ( K 1 ) ⊗ � CFK ( K 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; , ∇ K ( t ) = t 2 − 1 + t − 2 e. g. for K = Connected sum formula: ∇ K 1 # K 2 ( t ) = ∇ K 1 ( t ) · ∇ K 2 ( t ) � HFK ( K ) a doubly graded f. g. Abelian group, invariant of oriented knots and links; Connected sum formula: � CFK ( K 1 # K 2 ) = � CFK ( K 1 ) ⊗ � CFK ( K 2 ) HFK ( K )) := � ( − 1) h rk( � χ ( � HFK h , a ) t a = ∇ K ( t ) Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 3 / 10

Motivation knot Floer homology c a t e g o r i fi e s Alexander polynomial Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Motivation knot Floer homology Khovanov homology c c a a t t e e g g o o r r i i fi fi e e s s Alexander polynomial Jones polynomial Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Motivation Bar-Natan’s Khovanov homology for tangles categorifies knot Floer homology Khovanov homology c c a a t t Jones polynomial e e g g o o for tangles r r i i fi fi e e s s Alexander polynomial Jones polynomial Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

Motivation Bar-Natan’s Khovanov homology for tangles categorifies knot Floer homology Khovanov homology c c a a t t Jones polynomial e e g g ? o o for tangles r r i i fi fi e e s s Alexander polynomial Jones polynomial Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 4 / 10

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

b b Definition of ∇ K for knots and links Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

b b b b b Definition of ∇ K for knots and links 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 occupied by exactly one marker. a Kauffman state Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

b b b b b Definition of ∇ K for knots and links 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 occupied by exactly one marker. a Kauffman state Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

b b b b b Definition of ∇ K for knots and links 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 1 that each closed region is occupied by exactly one t marker. t t 1 1 a labelled Kauffman state − t − 1 Alexander Code for a positive crossing Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

b b b b b Definition of ∇ K for knots and links 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 1 that each closed region is occupied by exactly one t marker. t t 1 1 ∇ K ( t ) = t 2 + − t − 1 Alexander Code for a positive crossing Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

b b b b b Definition of ∇ K for knots and links 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 − t − 1 occupied by exactly one marker. 1 t t 1 1 ∇ K ( t ) = t 2 − 1 + − t − 1 Alexander Code for a positive crossing Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

b b b b b Definition of ∇ K for knots and links 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 − t − 1 occupied by exactly one marker. − t − 1 t 1 1 1 ∇ K ( t ) = t 2 − 1 + t − 2 − t − 1 Alexander Code for a positive crossing Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 5 / 10

b b Definition of ∇ K for knots and links 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 occupied by exactly one marker. t 1 1 ∇ K ( t ) = t 2 − 1 + t − 2 − t − 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

b b b b Definition of ∇ s T for tangles Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

b b b b b b b b b Definition of ∇ s T for tangles Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10

b b b b b b b b b Definition of ∇ s T for tangles Definition A generalised Kauffman state of a tangle diagram T is an assignment of a marker to one of the four regions at each crossing such that each closed region is occupied by exactly one marker, Claudius Zibrowius (Cambridge) Alexanderpolynomials for Tangles ECSTATIC 2015 6 / 10