Rational surgery exact triangles in Heegaard Floer homology

TL;DR

This work constructs a broad family of surgery exact triangles in Heegaard Floer theory for all positive rational slopes $p/q$, parameterized by triples $(p,q,k)$ with $p,q$ coprime and $0\le k\le p-1$. The key strategy reduces the global problem to a local, combinatorial calculation on a genus-1 Heegaard diagram with attaching curves carrying carefully chosen local systems, together with a triangle-detection framework. A novel invariant, $\boldsymbol{HFK}_{p,q,k}^{-}(Y,K)$, acts as a $2k/p$-modified knot Floer homology with twisted coefficients, unifying and extending the classical Ozsváth–Szabó $n$- and $1/n$-surgery triangles and the $2$-surgery triangle. The main local computation verifies the existence of cycles yielding vanishing triangle maps and identity-like quadrilateral maps modulo $U$, enabling neck-stretching arguments to produce the full exact triangles; the paper also develops a detailed hat/minus setup via covers and zig-zags and provides explicit low-rank examples with standard invariants. These results open avenues for instanton-Floer analogues and deepen the understanding of rational-slope behavior in Heegaard Floer theory, with potential applications to knot concordance and 3-manifold topology.

Abstract

We construct a new family of surgery exact triangles in Heegaard Floer theory over the field with two elements. This family generalizes both Ozsváth and Szabó's $n$- and $1/n$-surgery exact triangles for positive integers $n$ and the author's recent 2-surgery exact triangle to all positive rational slopes. The construction reduces to a combinatorial problem that involves triangle and quadrilateral counting maps in a genus 1 Heegaard diagram. The main contribution of this paper is solving this combinatorial problem, which is particularly tricky for slopes $r\neq n,1/n$; one key idea is to use an involution that is closely related to the ${\rm Spin}^{c}$ conjugation symmetry.

Figures (15)

• Figure 1.1: A genus $1$ Heegaard diagram $(\mathbb{T}^{2},\beta_{0},\beta_{r},\beta_{\infty},z)$ for $r=5/3$. The intersection points $\{\theta_{0},\theta_{1},\theta_{2},\theta_{3},\theta_{4}\}=\beta_{0}\cap\beta_{r}$, $\{\xi_{0},\xi_{1},\xi_{2}\}=\beta_{r}\cap\beta_{\infty}$, and $\{\zeta\}=\beta_{\infty}\cap\beta_{0}$ are labelled.
• Figure 2.1: A local diagram for the Heegaard diagram near $A$ for Lemma \ref{['lem:local-system-simple-lemma']}. If a path $\gamma$ in $\boldsymbol{\beta}$ goes from top to bottom in this diagram, then $\#(A\cap\gamma)=1$.
• Figure 2.2: (a), (b): Two cases for a standard translate $\boldsymbol{\alpha}'$ of $\boldsymbol{\alpha}$, near the oriented arc $A_{\boldsymbol{\alpha}}$. The basepoint is not drawn. (c): $x'$ is the nearest point to $x$.
• Figure 3.1: Curves and intersection points on $\mathbb{T}^{2}$ for $(p,q)=(5,3)$. Right: zoomed in to show the intersection points $\Theta_{0}^{+},\Theta_{r}^{+},\Theta_{\infty}^{+}$ and the oriented arcs $A_{\infty},A_{0}$. If a path $\gamma$ in $\beta_{\infty}$ goes from top to bottom, then $\#(A_{\infty}\cap\gamma)=1$, and if a path $\gamma$ in $\beta_{0}$ goes from left to right, then $\#(A_{0}\cap\gamma)=1$.
• Figure 4.1: Some examples for the cover $\widetilde{\mathbb{T}^{2}}$ of the genus $1$ Heegaard diagram. See Subsection \ref{['subsec:An-explicit-formula']} for the meaning of $\boldsymbol{\theta}_{i,\ell}$, $\boldsymbol{\xi}_{j}$, and $\boldsymbol{\zeta}_{i,\ell}$. The intersection point labelled $\boldsymbol{\zeta}_{0,0}$ is the origin. In all three cases, the lifts of standard basis elements are labelled, and the set $\widetilde{z(u)}$ consists of the points $z$ in the figure. (a): $(p,q,k)=(5,3,0)$, $u=(1,1,1,1,1)$, $c=(1,1,1,0,0)$. (b): $(p,q,k)=(5,3,2)$, $u=(1,1,U,1,U)$, $c=(0,0,1,1,1)$. (c): $(p,q,k)=(3,5,0)$, $u=(1,1,1)$, $c=(2,2,1)$.

Theorems & Definitions (134)

• Theorem 1.1: Main theorem, special case $k=0$
• Remark 1.2
• Theorem 1.3: Main theorem, general case
• Remark 1.4: Theorems \ref{['thm:rational-surgery-k=00003D0']} and \ref{['thm:rational-surgery-kgeneral']} for the hat, plus, and infinity versions
• Remark 1.6: Necessity of local systems
• Definition 1.8: Knot Floer chain complex MR2065507MR2704683
• Definition 1.9: Heegaard Floer chain complex with $\mathbb{F}[\mathbb{Z}/N\mathbb{Z}]$-twisted coefficients MR2113020
• Definition 1.10: $t$-modified knot Floer chain complex 1407.1795
• Remark 1.11
• Definition 1.12: Modified knot Floer chain complex $\boldsymbol{CFK}_{p,q,k}^{-}(Y,K)$