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Absolute $\mathbb{Z}/2$ gradings in real Heegaard Floer homology

Eha Srivastava

Abstract

Real Heegaard Floer homology is an invariant associated to a three-manifold equipped with an involution with nonempty fixed set of codimension two. We show that when the image of the fixed point set is nullhomologous in the quotient, the real Heegaard Floer homology groups admit an absolute $\mathbb{Z}/2$ grading; in particular this applies to double branched covers of links in $S^3$. As an application, we define a $\mathbb{Z}$-valued invariant of knots, which is the appropriate signed analogue of Miyazawa's degree invariant. Furthermore, we show that this invariant is equal to the Alexander polynomial of the knot evaluated at $i$.

Absolute $\mathbb{Z}/2$ gradings in real Heegaard Floer homology

Abstract

Real Heegaard Floer homology is an invariant associated to a three-manifold equipped with an involution with nonempty fixed set of codimension two. We show that when the image of the fixed point set is nullhomologous in the quotient, the real Heegaard Floer homology groups admit an absolute grading; in particular this applies to double branched covers of links in . As an application, we define a -valued invariant of knots, which is the appropriate signed analogue of Miyazawa's degree invariant. Furthermore, we show that this invariant is equal to the Alexander polynomial of the knot evaluated at .
Paper Structure (10 sections, 13 theorems, 78 equations, 11 figures)

This paper contains 10 sections, 13 theorems, 78 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The absolute Z/2 grading for Heegaard Floer homology
  3. The definition of the grading
  4. Relation to gr
  5. The absolute Z/2 grading for real Heegaard Floer homology
  6. Background on real Heegaard Floer homology
  7. The definition of the grading
  8. The Euler Characteristic
  9. The sign of the Euler characteristic
  10. Double branched covers of links

Key Result

Theorem 1.1

Let $Y$ be a closed oriented three-manifold, and let $(\Sigma, \boldsymbol{\alpha}, \boldsymbol{\beta}, z)$ be an admissible pointed Heegaard diagram for $Y$. The function $\operatorname{gr}' : \mathit{CF}^\infty(\mathbb{T}_\alpha, \mathbb{T}_\beta) \to \mathbb{Z}/2$ determines a well-defined absolu

Figures (11)

  • Figure 1: An example of a pair of additional alpha and beta curves.
  • Figure 2: Adding a small unknotted tube to $F_{i-1}$.
  • Figure 3: Additional tubes coming from a subdivision of the triangulation. In this example, the bold edge in the bottom left is already contained in $\Gamma$. Adding the new tubes can be viewed as a sequence of three local stabilizations.
  • Figure 4: A triangulation of $H$. The black bold edges are part of $\Gamma$ and the purple edges are $\Gamma' \setminus \Gamma$, namely the cores of the two bands forming $H$.
  • Figure 5: The surfaces $F_T$ and $F_T'$ around $H$.
  • ...and 6 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 1.5
  • Proposition 1.6
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • ...and 20 more