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Localization in Khovanov homology

Matthew Stoffregen, Melissa Zhang

Abstract

We construct equivariant Khovanov spectra for periodic links, using the Burnside functor construction introduced by Lawson, Lipshitz, and Sarkar. By identifying the fixed-point sets, we obtain rank inequalities for odd and even Khovanov homologies, and their annular filtrations, for prime-periodic links in $S^3$.

Localization in Khovanov homology

Abstract

We construct equivariant Khovanov spectra for periodic links, using the Burnside functor construction introduced by Lawson, Lipshitz, and Sarkar. By identifying the fixed-point sets, we obtain rank inequalities for odd and even Khovanov homologies, and their annular filtrations, for prime-periodic links in .

Paper Structure

This paper contains 35 sections, 44 theorems, 163 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Techniques
  5. Khovanov homologies and periodic links
  6. The cube category
  7. Even Khovanov homology $\mathit{Kh}$
  8. Homology from functors
  9. Odd Khovanov homology $\mathit{Kh}_o$
  10. Annular filtrations
  11. Periodic links
  12. Periodic links and Khovanov homologies
  13. Burnside categories and functors
  14. The Burnside category
  15. Decorated Burnside categories
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\tilde{L}$ be a $p^n$-periodic link, for a prime $p$, with quotient link $L$. Let $\mathit{Kh}(\tilde{L};\mathbb{F}_p)$ (resp. $\mathit{Kh}_o(\tilde{L};\mathbb{F}_p)$) denote the Khovanov homology (resp. odd Khovanov homology) of $\tilde{L}$, with coefficients in $\mathbb{F}_p$, the field of

Figures (9)

  • Figure 2.1: The six types of saddle interactions between circles in the annulus. Components of the (even) Khovanov differential are listed in the side columns, with components that fail to preserve the annular $(k)$-grading in parentheses; these decrease $\mathrm{gr}_k$ by exactly $-2$. For the odd case, the signs may differ, depending on context.
  • Figure 6.1: An example Burnside functor $F\colon \underline{2}^1 \to \mathscr{B}_{\mathbb{Z}_2}$. We visualize elements of $F(1),F(0)$ as dots, and regard the morphism $F(\phi_{1,0})$ as a collection of arrows. Here, we let $F(1)=\{1,x_1,x_2,x_1x_2\}$, the set of Khovanov generators associated to a resolution configuration of two circles, and $F(0)=\{1,y_1\}$, the set of Khovanov generators associated to a single circle. Set $F(\phi_{1,0}) = \{a_1, a_2, a_3\}$; $s(a_i)$ is given by the tail of the arrow $a_i$, and $t(a_i)$ is given by the head of the arrow $a_i$. This is the Khovanov-Burnside functor associated to two circles merging to a single circle. Generators at the same height have the same quantum grading.
  • Figure 6.2: The three equivariant annular merges, with $p = 5$ illustrated. Here, $\Delta$ stands for 'split' and $m$ stands for 'merge.'
  • Figure 6.3: The $\mathbb{V}\otimes \mathbb{V}\to \mathbb{W}$ case for $p=5$. The map downstairs is $x_1,x_2\to x$, where $x_1,x_2\in Z(D_0)$, $x\in Z(D_1)$. Let $Z(\tilde{D}_{0^5})=\{\tilde{x}_1,\tilde{x}_2\}$, the elements over $x_1,x_2$, and let $Z(\tilde{D}_{1^5})=\{y_1,\dots,y_5\}$, related by the action of $\mathbb{Z}_5$. Then upstairs the map on fixed points is $\tilde{x}_i\to (y_5-y_1)(y_1-y_2)(y_2-y_3)(y_3-y_4)y_5=y_1y_2y_3y_4y_5$. Moreover, the element $1\in \mathfrak{F}_o'(0^5)$ is sent to a term in $\mathfrak{F}_o'(1^5)$ which is killed by projection to the summand of invariant generators, and the element $\tilde{x}_1\tilde{x}_2$ is annihilated by $\mathfrak{F}_o'(\phi^{\mathrm{op}}_{0^5,1^5})$. This verifies Lemma \ref{['lem:fake-functor']} in this example.
  • Figure 6.4: The annular resolution configurations of type X.
  • ...and 4 more figures

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2: Definition 2.1 lshomotopytype
  • Remark 2.4
  • Proposition 2.6: cf. z-annular-rank
  • proof
  • Definition 3.3
  • Lemma 3.4: oddkh
  • Definition 3.5
  • ...and 93 more