Khovanov invariants for knots

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Khovanov invariants for knots

Maciej Borodzik

Institute of Mathematics, University of Warsaw

Warsaw, 2018

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Knots

A knot is a possibly tangled circle inR³:

Maciej Borodzik Khovanov invariants for knots

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Knots

A knot is a possibly tangled circle inR³: Definition

A knot inR³is an image of a smooth embeddingφ : S¹→ R³. A link is “a knot with more than one component”.

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Definition

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Knots

A knot is a possibly tangled circle inR³: Definition

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Definition

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Distinguishing knots

A knot invariant assigns a simpler (more tractable) object to a knot;

Should be the same no matter how the knot is drawn; Should be computable;

Should have a meaning; Should really distinguish knots.

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Distinguishing knots

Should be the same no matter how the knot is drawn;

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Distinguishing knots

Should be computable;

Should have a meaning; Should really distinguish knots.

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Distinguishing knots

Should have a meaning;

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Distinguishing knots

Should have a meaning;

Should really distinguish knots.

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Polynomial invariants

Assign a polynomial to a knot.

Ocneanu, Millet, Freyd, Yetter, and independently by Przytycki and Traczyk in 1986.

Alexander and Jones polynomials are polynomials in one variable (formally int^1/2andt^−1/2, so Laurent polynomials. HOMLYPT is a two-variable polynomial.

There are many more polynomial invariants, but these are the most basic. They have a special property.

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Polynomial invariants

Alexander polynomialdefined in 1928.

Jones polynomial discovered in 1984.

HOMFLYPT polynomial constructed in 1985 by Hoste, Ocneanu, Millet, Freyd, Yetter, and independently by Przytycki and Traczyk in 1986.

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Polynomial invariants

Alexander polynomial defined in 1928.

Jones polynomialdiscovered in 1984.

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Polynomial invariants

HOMFLYPT polynomialconstructed in 1985 by Hoste, Ocneanu, Millet, Freyd, Yetter, and independently by Przytycki and Traczyk in 1986.

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Polynomial invariants

Alexander and Jones polynomials are polynomials in one variable (formally int^1/2andt^−1/2, so Laurent polynomials.

HOMLYPT is a two-variable polynomial.

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Polynomial invariants

Alexander and Jones polynomials are polynomials in one variable (formally int^1/2andt^−1/2, so Laurent polynomials.

HOMLYPT is a two-variable polynomial.

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Skein relation

L+ L₀ L₋

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Skein relation

L+ L₀ L₋

Definition (Informal)

A skein relation is a relation between the polynomials for links differing at a single place of the diagram.

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Skein relation for Alexander and Jones polynomial

LetA be the Alexander polynomial andJ be the Jones polynomial.

t J_L₊(t) − tJ_L₋(t) = (t^1/2− t^−1/2)J_L₀(t).

Remark

There are various normalizations of the Alexander and Jones

polynomials, which lead to different looking formulas.

L+ L₀ L−

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Skein relation for Alexander and Jones polynomial

We have: A_L₊(t) − A_L₋(t) = (t^1/2− t^−1/2)A_L₀(t).

For Jones:

t⁻¹J_L₊(t) − tJ_L₋(t) = (t^1/2− t^−1/2)J_L₀(t).

Remark

L+ L₀ L−

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Skein relation for Alexander and Jones polynomial

For Jones:

t⁻¹J_L₊(t) − tJ_L₋(t) = (t^1/2− t^−1/2)J_L₀(t).

looking formulas.

L+ L₀ L−

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Skein relation for Alexander and Jones polynomial

For Jones:

t⁻¹J_L₊(t) − tJ_L₋(t) = (t^1/2− t^−1/2)J_L₀(t).

Remark

L+ L₀ L−

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Alexander polynomial Jones polynomial

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Jones vs. Alexander

Alexander polynomial Jones polynomial Multiplicative for connected

sums

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sums

as well

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Jones vs. Alexander

sums

as well Belongs toZ[t , t⁻¹]for knots as well

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sums

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Jones vs. Alexander

sums

as well Belongs toZ[t , t⁻¹]for knots as well SatisfiesA(t⁻¹) =A(t)

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sums

SatisfiesA(t⁻¹) =A(t) No such relation

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Jones vs. Alexander

sums

∆K(t) = ±1for knots

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sums

∆K(t) = ±1for knots Jdetermined on roots of unity of order2, 3, 4

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Jones vs. Alexander

Alexander polynomial Jones polynomial Belongs toZ[t , t⁻¹]for knots as well

All polynomials satisfying above points can be realized

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Alexander polynomial Jones polynomial Belongs toZ[t , t⁻¹]for knots as well

??

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Jones vs. Alexander

Alexander polynomial Jones polynomial SatisfiesA(t⁻¹) =A(t) No such relation

∆_K(t) = ±1for knots Jdetermined on roots of unity of order2, 3, 4

??

There are knots withA(t) ≡ 1

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Alexander polynomial Jones polynomial SatisfiesA(t⁻¹) =A(t) No such relation

??

There are knots withA(t) ≡ 1 ??

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Jones vs. Alexander

??

Topological meaning per- fectly understood

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??

We know very little beyond combinatorics

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Jones vs. Alexander

??

Computable in polynomial time

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??

Computable in polynomial time

Most likely exponential time needed

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Cube of resolutions. Part 1.

We specify resolutions of a knot diagram.

Take a knot.

Enumerate its crossings.

Any triple{0, 1}³ gives a resolution.

1 0

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Cube of resolutions. Part 1.

Take a knot.

1 0

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Cube of resolutions. Part 1.

Take a knot.

Enumerate its crossings. Any triple{0, 1}³ gives a resolution.

1 0

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Cube of resolutions. Part 1.

Take a knot.

1 0

1

2 3

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Cube of resolutions. Part 1.

Take a knot.

0-resolution of the first crossing.

1 0

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Cube of resolutions. Part 1.

Take a knot.

1-resolution of the first crossing.

1 0

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Cube of resolutions. Part 1.

Take a knot.

0-resolution of the second crossing.

1 0

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Cube of resolutions. Part 1.

Take a knot.

010resolution.

1 0

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Cube of resolutions. Part 1.

Take a knot.

010resolution.

1 0

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000

3circles

111

2circles 2circles

010

2circles 100

2circles

1circle 101

1circle 110

1circle

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Cube of resolution

000

3circles

111

2circles 001

2circles 010

2circles 100

2circles

011

1circle 101

1circle 110

1circle

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000

(q + q⁻¹)³

111

(q + q⁻¹)² (q + q⁻¹)²

010

(q + q⁻¹)² 100

(q + q⁻¹)²

(q + q⁻¹) 101

(q + q⁻¹) 110

(q + q⁻¹)

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Cube of resolution

000

(q + q⁻¹)³

111

(q + q⁻¹)² 001

(q + q⁻¹)² 010

(q + q⁻¹)² 100

(q + q⁻¹)²

011

(q + q⁻¹) 101

(q + q⁻¹) 110

(q + q⁻¹)

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Cube of resolution

000

(q + q⁻¹)³

111

(q + q⁻¹)² 001

(q + q⁻¹)² 010

(q + q⁻¹)² 100

(q + q⁻¹)²

011

(q + q⁻¹) 101

(q + q⁻¹) 110

(q + q⁻¹)

(q + q⁻¹)³ 3(q + q⁻¹)² 3(q + q⁻¹) (q + q⁻¹)²

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Cube of resolution

000

(q + q⁻¹)³

111

(q + q⁻¹)² 001

(q + q⁻¹)² 010

(q + q⁻¹)² 100

(q + q⁻¹)²

011

(q + q⁻¹) 101

(q + q⁻¹) 110

(q + q⁻¹)

(q + q⁻¹)³q⁰ 3(q + q⁻¹)²q¹ 3(q + q⁻¹)q² (q + q⁻¹)²q³

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000

(q + q⁻¹)³

111

(q + q⁻¹)² (q + q⁻¹)²

010

(q + q⁻¹)² 100

(q + q⁻¹)²

(q + q⁻¹) 101

(q + q⁻¹) 110

(q + q⁻¹)

(q + q⁻¹)³q⁰ -3(q + q⁻¹)²q¹ + 3(q + q⁻¹)q² - (q + q⁻¹)²q³

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Jones polynomial

We have

(q⁻¹+q)³− 3q(q⁻¹+q)²+3q²(q⁻¹+q) − q³(q⁻¹+q) =

− q⁶(q⁻²− q⁻³+q⁻⁴− q⁻⁹)

In this way we obtain the Jones polynomial for the (negative) trefoil. Factor−q⁻⁶is a normalization.

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We have

(q⁻¹+q)³− 3q(q⁻¹+q)²+3q²(q⁻¹+q) − q³(q⁻¹+q) =

− q⁶(q⁻²− q⁻³+q⁻⁴− q⁻⁹) In this way we obtain the Jones polynomial for the (negative) trefoil. Factor−q⁻⁶is a normalization.

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Khovanov’s approach

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Main Idea

Replace factorq + q⁻¹in the cube of resolution by a two-dimensional vector spaceV.

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Khovanov’s approach

000

3circles

111

2circles 001

2circles 010

2circles 100

2circles

011

1circle 101

1circle 110

1circle

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000

V³

111

V² V²

010

V² 100

V²

V 101

V 110

V

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Khovanov’s approach

Explanation

The meaning ofV³is the tensor product. An element inV³is a linear combination of triples(a, b, c)(written usuallya ⊗ b ⊗ c).

We havea₁⊗ b ⊗ c + a2⊗ b ⊗ c = (a1+a₂) ⊗b ⊗ c, but not a₁⊗ b₁⊗ c₁+a₂⊗ b₂⊗ c₂= (a₁+a₂) ⊗ (b₁+b₂) ⊗ (c₁+c₂).

dimV^⊗3= (dimV )³and not3 dim V!

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000

V³

111

V² V²

010

V² 100

V²

V 101

V 110

V

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Khovanov’s approach

000

V³

111

V² 001

V² 010

V² 100

V²

011

V 101

V 110

V

V^⊗3 V^⊗2⊕ V^⊗2⊕ V^⊗2 V ⊕ V ⊕ V V^⊗2

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000

V³

111

V² V²

010

V² 100

V²

V 101

V 110

V

V^⊗3 ? V^⊗2⊕ V^⊗2⊕ V^⊗2 ? V ⊕ V ⊕ V ? V^⊗2

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Khovanov’s approach

000

V³

111

V² 001

V² 010

V² 100

V²

011

V 101

V 110

V

V^⊗3 ? V^⊗2⊕ V^⊗2⊕ V^⊗2 ? V ⊕ V ⊕ V ? V^⊗2

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Maps in Khovanov’s approach

An arrow can either merge two circles into one.

Without extra structure, it is hard to define such maps consistently.

010 011

merge

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Maps in Khovanov’s approach

Or split one into two circles.

In the first case we need a mapV ⊗ V → V. In the second case we need a mapV → V ⊗ V. Without extra structure, it is hard to define such maps consistently.

011 111

split

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Maps in Khovanov’s approach

In the first case we need a mapV ⊗ V → V.

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Maps in Khovanov’s approach

In the first case we need a mapV ⊗ V → V. In the second case we need a mapV → V ⊗ V.

Without extra structure, it is hard to define such maps consistently.

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In the first case we need a mapV ⊗ V → V. In the second case we need a mapV → V ⊗ V. Without extra structure, it is hard to define such maps consistently.

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Maps in Khovanov homology

Think ofV as a space of affine functionsax + bwith a, b ∈ Z.

The mapV ⊗ V → V is the linear part of the product: 1 ⊗ 1 7→ 1,x ⊗ 1, 1 ⊗ x 7→ x , x ⊗ x 7→ 0.

The map fromV → V ⊗ V ‘copies’ the function on the generators: x 7→ x ⊗ x,1 7→ 1 ⊗ 1.

Combining these maps (and after some sign adjustments) we obtain maps replacing+and−signs.

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Maps in Khovanov homology

The mapV ⊗ V → V is the linear part of the product:

1 ⊗ 1 7→ 1,x ⊗ 1, 1 ⊗ x 7→ x , x ⊗ x 7→ 0.

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Maps in Khovanov homology

1 ⊗ 1 7→ 1,x ⊗ 1, 1 ⊗ x 7→ x , x ⊗ x 7→ 0.

The map fromV → V ⊗ V ‘copies’ the function on the generators:x 7→ x ⊗ x,1 7→ 1 ⊗ 1.

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1 ⊗ 1 7→ 1,x ⊗ 1, 1 ⊗ x 7→ x , x ⊗ x 7→ 0.

The map fromV → V ⊗ V ‘copies’ the function on the generators:x 7→ x ⊗ x,1 7→ 1 ⊗ 1.

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Global maps. Revised

000

V^⊗3

111

V^⊗2 001

V^⊗2 010

V^⊗2 100

V^⊗2

011

V 101

V 110

V

V^⊗3 d₀V^⊗2⊕ V^⊗2⊕ V^⊗2 V ⊕ V ⊕ V V^⊗2

−→ −→

^d¹

−→

^d²

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Theorem (Khovanov 2000)

The mapsd₀,d₁andd₂satisfyd₂◦ d₁=0andd₁◦ d₀=0. The abelian groupskerd_i/ imd_i−1are independent of the knot diagram.

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Khovanov invariant

The mapsd₀,d₁andd₂satisfyd₂◦ d₁=0andd₁◦ d₀=0. The abelian groupskerd_i/ imd_i−1are independent of the knot diagram.

Remark

In mathematics, a sequence of vector spacesV₀, . . . ,V_s together with linear mapsd_i:V_i → V_i+1satisfyingd_i◦ d_i−1 =0 for alliis called acochain complex. The groupskerd_i/ imd_i−1 are calledcohomology groups.

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The mapsd₀,d₁andd₂satisfyd₂◦ d1=0andd₁◦ d0=0. The abelian groupskerd_i/ imd_i−1are independent of the knot diagram.

Remark

In mathematics, a sequence of vector spacesV₀, . . . ,Vs

together with linear mapsd_i:V_i → Vi+1satisfyingd_i◦ di−1 =0 for alliis called acochain complex. The groupskerd_i/ imd_i−1 are calledcohomology groups.

Yes, I know, saying ‘a vector space overZ’ is an abuse.

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Properties of Khovanov invariant

Detects the unknot (Kronheimer, Mrowka 2011).

Detects the Hopf link and the trefoil.

Specifies to and generalizes the Jones polynomial. Can be used to prove the Milnor’s conjecture (on the unknotting number of torus knots).

Computational complexity is daunting.

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Properties of Khovanov invariant

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Properties of Khovanov invariant

Specifies to and generalizes the Jones polynomial.

Can be used to prove the Milnor’s conjecture (on the unknotting number of torus knots).

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Properties of Khovanov invariant

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Properties of Khovanov invariant

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Making it better

We said that Khovanov invariant is cohomology of a chain complex.

Given a knotK can one construct a topological spaceX such that the cohomology ofX is the Khovanov invariant ofK? Is there a consistent construction?

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Making it better

Many people are familiar with cohomology of topological spaces.

Question

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Many people are familiar with cohomology of topological spaces.

Question

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Lipsitz and Sarkar construction

First construction of Khovanov homotopy type using flow categories and Cohen-Jones-Segal (2012).

New invariants of knots coming from cohomological operations (2013).

Another construction of flow categories using cubical flow categories and Burnside categories (2014, jointly with Lawson).

Invited to the ICM in 2018.

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Lipsitz and Sarkar construction

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Lipsitz and Sarkar construction

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Periodic knots

A knot isp-periodic if it admits a diagram invariant under rotation byZ/p.

Which knots arep-periodic?

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Periodic knots

periodic 3-periodic not periodic

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Periodic knots

periodic

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Periodic knots

periodic 3-periodic

not periodic

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periodic 3-periodic not periodic

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Equivariant Khovanov invariants

Politarczyk 2014: Construction of equivariant Khovanov invariants;

Politarczyk 2015: New periodicity criterion based on equivariant Jones polynomial;

Borodzik, Politarczyk 2018: Another, much stronger, periodicity criterion based on equivariant Khovanov invariants.

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Equivariant Khovanov invariants

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Equivariant Khovanov invariants

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Question

Does there exists equivariant Khovanov homotopy type?

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Equivariant Khovanov homotopy type

Theorem (B. — Politarczyk — Silvero 2018, Stoffregen — Zhang 2018)

There exists equivariant Khovanov homotopy type.

BPS approach proves also that equivariant cohomology of this space is Politarczyk’s equivariant Khovanov invariant.

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Theorem (B. — Politarczyk — Silvero 2018, Stoffregen — Zhang 2018)

There exists equivariant Khovanov homotopy type.

BPS approach proves also that equivariant cohomology of this space is Politarczyk’s equivariant Khovanov invariant.

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Perspectives

Construct HOMLYPT homotopy type;

Construct a homotopy type that reflects and intertwines the quantum grading.

Understand, why Khovanov invariants work.

Find a simpler way to calculate Khovanov invariants.

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Perspectives

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Perspectives

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