Knot Floer homology of positive braids

TL;DR

The paper addresses determining the next-to-top term of knot Floer homology for positive braid links. It develops a skein- and Seifert-graph–driven framework to compute $\widehat{HFK}(L,g-1)$, culminating in a closed formula $\widehat{HFK}(L,g-1) \cong \mathbb{F}^{p(L)+|L|-s(L)}[-1] \otimes (\mathbb{F}[0]\oplus \mathbb{F}[-1])^{\otimes s(L)-1}$ for positive braid links, with a key corollary for prime positive braid knots: $\widehat{HFK}(K,g-1) \cong \mathbb{F}[-1]$ and a specific symmetric form for $\Delta_K(t)$. The work connects to broader themes involving L-space and diagonal knots, and provides concrete examples (the families $L_n$ and $R_n$) to illustrate the range of possible next-to-top terms. It also proposes a conjecture that prime fibered positive knots satisfy $\widehat{HFK}_{-1}(K,g-1) \cong \mathbb{F}$, guiding future investigations into the interaction between positivity, fiberedness, and Heegaard Floer invariants.

Abstract

We compute the next-to-top term of knot Floer homology for positive braid links. The rank is 1 for any prime positive braid knot. We give some examples of fibered positive links that are not positive braids.

Figures (6)

• Figure 1: The skein relation, we have $L_+$, $L_-$ and $L_0$ from left to right.
• Figure 2: The two crossings are adjacent if there is no other crossing in the thickened segments.
• Figure 3: This is $\sigma_1^2\sigma_2^3\sigma_1\sigma_2^4$ representing the knot $K_{10_{139}}$. The filled box stands for the marked point on the knot (so no Kauffman state can have coordinate in the regions adjacent to the box). The Maslov-Alexander bigradings for the Kauffman state at black dots, white dots, crosses and boxes (adding a black dot as coordinate whenever we encounter a crossing with no corresponding symbol) are $(0,4)$, $(-1,3)$, $(-1,3)$, and $(0,3)$, respectively.
• Figure 4: The figure at the top is the Seifert surface of $L_2$ and the bottom is a part of the Seifert surface of $L_n$.
• Figure 5: Links in the left column are $K_{10_{154}}$, $L_{9n_9}$, and $L_{8n_6}$ while links in the right column are $K_{10_{161}}$, and $L_{9n_{19}}$. Links are chosen with specific orientations by arrows. Red circles are the adjacent crossings that we resolve to obtain the links below them. Figures are from Knotinfo.

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Conjecture 1.4
• Theorem 2.1: Cromwell
• Proposition 2.2
• Proposition 3.1: OSes
• Proposition 3.2: Stallings, OScontact
• Remark 3.3
• Corollary 3.4