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A new concordance homomorphism from Khovanov homology

Lukas Lewark

Abstract

The universal Khovanov chain complex of a knot modulo an appropriate equivalence relation is shown to yield a homomorphism on the smooth concordance group, which is strictly stronger than all Rasmussen invariants over fields of different characteristics combined.

A new concordance homomorphism from Khovanov homology

Abstract

The universal Khovanov chain complex of a knot modulo an appropriate equivalence relation is shown to yield a homomorphism on the smooth concordance group, which is strictly stronger than all Rasmussen invariants over fields of different characteristics combined.
Paper Structure (8 sections, 18 theorems, 46 equations, 1 figure)

This paper contains 8 sections, 18 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on graded rings
  3. Khovanov homology as a black box
  4. A glimpse into the black box
  5. Rasmussen invariants over fields
  6. The new homomorphism S
  7. Calculations of S
  8. Further properties of S

Key Result

Theorem 1.1

Modulo $Z$-- equivalence, knot-like chain complexes form an abelian group $\mathcal{G}$ with group operation the tensor product, the inverse of a complex given by its dual, and the complex with graded rank 1 as neutral element. Associating to a knot $K$ the $Z$-- equivalence class of the universal K

Figures (1)

  • Figure 1: The first two knots in \ref{['conj:whitehead']}: $W^+_3(T_{2,3})$ and $W^+_6(T_{2,5})$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Example 2.1
  • Definition 3.1
  • Proposition 5.1
  • lemma 5.2
  • proof
  • proof : Proof of \ref{['prop:structureF']}
  • ...and 38 more