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Bar-Natan Homology for null homologous links in \mathbb{RP}^3

Daren Chen

TL;DR

Bar-Natan homology for null-homologous links in $\mathbb{RP}^{3}$ over $\mathbb{F}_2$ is constructed as a deformation of the Khovanov complex, using twisted orientations to define a canonical basis and a corresponding $s$-invariant $s^{BN}_{\mathbb{RP}^{3}}$. The authors establish a cobordism formalism via twisted orientable cobordisms and maps of degree $\chi(\Sigma)$, yielding a genus bound for twisted orientable slice surfaces. They show the invariant detects differences from the Lee-based $s$-invariant and provide explicit bounds and examples illustrating the distinct information carried by Bar-Natan theory on $\mathbb{RP}^{3}$. The work suggests avenues toward equivariant/double-cover refinements and extensions to broader 3-manifold settings.

Abstract

In this paper, we introduce Bar-Natan homology for null homologous links in \mathbb{RP}^3 over the field of two elements. It is a deformation of the Khovanov homology in \mathbb{RP}^3 defined by Asaeda, Przytycki and Sikora. We also define an s-invariant from this deformation using the same recipe as for links in S^3, and prove some genus bound using it. The key ingredient is the notion of twisted orientation for null homologous links and cobordisms in \mathbb{RP}^3.

Bar-Natan Homology for null homologous links in \mathbb{RP}^3

TL;DR

Bar-Natan homology for null-homologous links in over is constructed as a deformation of the Khovanov complex, using twisted orientations to define a canonical basis and a corresponding -invariant . The authors establish a cobordism formalism via twisted orientable cobordisms and maps of degree , yielding a genus bound for twisted orientable slice surfaces. They show the invariant detects differences from the Lee-based -invariant and provide explicit bounds and examples illustrating the distinct information carried by Bar-Natan theory on . The work suggests avenues toward equivariant/double-cover refinements and extensions to broader 3-manifold settings.

Abstract

In this paper, we introduce Bar-Natan homology for null homologous links in \mathbb{RP}^3 over the field of two elements. It is a deformation of the Khovanov homology in \mathbb{RP}^3 defined by Asaeda, Przytycki and Sikora. We also define an s-invariant from this deformation using the same recipe as for links in S^3, and prove some genus bound using it. The key ingredient is the notion of twisted orientation for null homologous links and cobordisms in \mathbb{RP}^3.
Paper Structure (5 sections, 19 theorems, 80 equations, 19 figures)

This paper contains 5 sections, 19 theorems, 80 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Bar-Natan homology for null homologous links in $\mathbb{RP}^{3}$
  3. Cobordism maps on Bar-Natan homology in $\mathbb{RP}^{3}$
  4. The $s$-invariant and genus bound
  5. Further directions

Key Result

Theorem 1.1

For a twisted oriented null homologous link $L \subset \mathbb{RP}^{3}$, one can associate a Bar-Natan chain complex $\mathit{CBN}(L)$ over the base field $\mathbb{F} = \mathbb{F}_2$, whose homology $\mathit{HBN}(L)$ is an invariant of twisted oriented $L$ as a bigraded vector space. More specifical and there is a basis of $\mathit{HBN}(L)$ given by $\left\{s_o\mid o \text{ is a twisted orientatio

Figures (19)

  • Figure 1: An example of difference in $s$-invariants defined using Lee deformation and Bar-Natan deformation
  • Figure 2: $0$ and $1$- Smoothings
  • Figure 3: $2$-$1$, $1$-$2$ and $1$-$1$ bifurcations
  • Figure 4: Singular graphs in $\mathbb{RP}^{2}$ with 2 singular points. See MR3189291
  • Figure 5: One example of a square in the cube of resolution
  • ...and 14 more figures

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 46 more