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Some link homologies in $ \mathbb{RP}^3 $

William Rushworth

Abstract

We introduce extensions of Khovanov homology and the Lee and Bar-Natan spectral sequences for links in $ \mathbb{RP}^3 $. These extensions are distinct to those previously defined by Asaeda-Przytycki-Sikora (and Gabrovšek's generalization), Chen, and Manolescu-Willis. The new Lee and Bar-Natan theories each yield Rasmussen invariants (that are distinct to one another). The invariant extracted from the new Lee homology is distinct to that defined by Manolescu-Willis; it is unclear if the same is true for the new Bar-Natan homology and that defined by Chen.

Some link homologies in $ \mathbb{RP}^3 $

Abstract

We introduce extensions of Khovanov homology and the Lee and Bar-Natan spectral sequences for links in . These extensions are distinct to those previously defined by Asaeda-Przytycki-Sikora (and Gabrovšek's generalization), Chen, and Manolescu-Willis. The new Lee and Bar-Natan theories each yield Rasmussen invariants (that are distinct to one another). The invariant extracted from the new Lee homology is distinct to that defined by Manolescu-Willis; it is unclear if the same is true for the new Bar-Natan homology and that defined by Chen.
Paper Structure (9 sections, 24 theorems, 57 equations, 3 figures)

This paper contains 9 sections, 24 theorems, 57 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extending Khovanov homology
  4. Extending Lee homology
  5. Extending Bar-Natan homology
  6. Functoriality
  7. Cobordism maps
  8. Rasmussen invariants
  9. Genus bounds

Key Result

Proposition 2.2

Two diagrams represent the same link if and only if they are related by a finite sequence of the classical Reidemeister moves and the following additional moves

Figures (3)

  • Figure 1: From left to right: a link diagram; a $2$-colouring of it; a diagram of a link that is not $2$-colourable.
  • Figure 2:
  • Figure 3: An oriented diagram associated to a twisted oriented diagram. In the twisted oriented diagram the two leftmost crossings are themselves twisted.

Theorems & Definitions (54)

  • Definition 2.1
  • Proposition 2.2: Drob90
  • Definition 2.3
  • Definition 2.4: $2$-colouring
  • Definition 2.5
  • Proposition 2.6
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 44 more