Combinatorial Description of Knot Floer Homology of Cyclic Branched Covers
J. Elisenda Grigsby
备注：
20 pages, 14 figures; Minor expositional improvements, typos corrected throughout (most seriously, the x,y coordinates used in discussion of intersection points beginning page 13 of previous version were incorrectly--but consistently--flipped)
We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks like a twisted version of the toroidal grid diagrams recently introduced by Manolescu, Ozsvath, and Sarkar. We conclude with a discussion of how one might obtain nice Heegaard diagrams for cyclic branched covers of more general knots.
Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic branched cover $Σ_m(K)$ obtained from a grid diagram for $K$, we give a combinatorial proof for the invariance of the associated combinatorial knot Floer homology over $\mathbb{Z}$.
Using a Heegaard diagram for the pullback of a knot $K \subset S^3$ in its cyclic double branched cover $Σ_2(K)$, we give a combinatorial proof for the invariance of knot Floer homology over $\mathbb{Z}$.
We prove that the rank of knot Floer homology detects the Hopf links, and generalize this result further to classify the links of the second smallest knot Floer homology. We also prove a knot Floer homology analog of arXiv:1910.04246v1 [math.GT] and give a partial answer to when the equality holds in the rank inequality between the knot Floer homology of a link and its sublinks.
A branched covering surface-knot is a surface-knot in the form of a branched covering over a surface-knot. For a branched covering surface-knot, we have a numerical invariant called the simplifying number. We show that branched covering surface-knots with degree three have the simplifying numbers less than three.
For a knot K and its knot Floer complex CFK^-(K), we introduce an algorithm to compute the bordered Floer bimodule of the complement of the knot and its meridian. The grading of the module computes spin^c-summands of a meridional knot in the large Dehn surgery manifold, which can be also extended to arbitrary framing n.
Given an equivariant knot $K$ of order $2$, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on $K$. This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of $S^{3}$ and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an appl…
Using the covering involution on the double branched cover of the three-sphere branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.
Let $\widetilde{K}$ be a 2-periodic knot in $S^3$ with quotient $K$. We prove a rank inequality between the knot Floer homology of $\widetilde{K}$ and the knot Floer homology of $K$ using a spectral sequence of Hendricks, Lipshitz and Sarkar. We also conjecture a filtered refinement of this inequality, for which we give computational evidence, and produce applications to the Alexander polynomials of $\widetilde{K}$ and $K$.
We provide criteria ensuring that a tunnel number one knot $K$ is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot $K'$ not equivalent to $K$.
We review some recent results in knot concordance and homology cobordism. The proofs rely on various forms of Heegaard Floer homology. We also discuss related open problems.
A well-known conjecture of Rasmussen states that for any knot $K$ in $S^{3}$, the rank of the reduced Khovanov homology of $K$ is greater than or equal to the rank of the reduced knot Floer homology of $K$. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture.
We prove that, up to local equivalences, a suitable truncation of the involutive knot Floer homology of a knot in $S^3$ and the involutive bordered Heegaard Floer theory of its complement determine each other. In particular, given two knots $K_1$ and $K_2$, we prove that the $\mathbb{F}_2[U,V]/(UV)$-coefficient involutive knot Floer homology of $K_1 \sharp -K_2$ is $ι_K$-locally trivial if $\widehat{CFD}(S^3 \backslash K_1)$ and $\widehat{CFD}(S^2 \backslash K_2)$ satisfy a certain condition which can be seen as the bordered counterpart of $ι_K$-local equivalence. We further establish an expli…
The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that $τ=g$ for each summand if and only if $τ=g$ holds for the Murasugi sum (with $τ$ and $g$ defined appropriately for multi-component links). Some applications are presented.
For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be homeomorphic to M.
We apply the theory of "peculiar modules" for the Floer homology of 4-ended tangles developed by Zibrowius (specifically, the immersed curve interpretation of the tangle invariants) to compute the Knot Floer Homology ($\widehat{HFK}$) of 3-stranded pretzel knots of the form ${P(2a,-2b-1,\pm(2c+1))}$ for positive integers $a,b,c$. This corrects a previous computation by Eftekhary; in particular, for the case of ${P(2a,-2b-1,2c+1)}$ where $b<c$ and $b<a-1$, it turns out the rank of $\widehat{HFK}$ is larger than that predicted by that work.
