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Invariants of surfaces in smooth 4-manifolds from link homology

Scott Morrison, Kevin Walker, Paul Wedrich

Abstract

We construct analogs of Khovanov-Jacobsson classes and the Rasmussen invariant for links in the boundary of any smooth oriented 4-manifold. The main tools are skein lasagna modules based on equivariant and deformed versions of $\mathfrak{gl}_N$ link homology, for which we prove non-vanishing and decomposition results.

Invariants of surfaces in smooth 4-manifolds from link homology

Abstract

We construct analogs of Khovanov-Jacobsson classes and the Rasmussen invariant for links in the boundary of any smooth oriented 4-manifold. The main tools are skein lasagna modules based on equivariant and deformed versions of link homology, for which we prove non-vanishing and decomposition results.
Paper Structure (10 sections, 15 theorems, 58 equations, 1 figure)

This paper contains 10 sections, 15 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equivariant and deformed skein modules
  3. Equivariant and deformed link homology
  4. Skein modules of 4-manifolds
  5. Skeins from surfaces
  6. Non-vanishing and genus bounds
  7. Decomposing deformed skein modules
  8. Relative second homology as link homology
  9. Non-vanishing modulo torsion
  10. Computations

Key Result

Theorem 2.1

Let $R$ be a commutative ring, optionally graded resp. filtered by an abelian group $\Gamma$. Given a functorial link homology theory for links in $\mathbb{R}^3$ with values in $R$, i.e. a functor: which additionally then $H$ extends to:

Figures (1)

  • Figure 1:

Theorems & Definitions (46)

  • Example
  • Remark
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3: Remark on grading conventions
  • Definition 2.4
  • Definition 2.5: Framing-changing input balls
  • Definition 2.6: Skein classes from surfaces
  • ...and 36 more