arXiv subjects

Fraser Binns

Publications and source records attributed to Fraser Binns.

14 recordsLinked to original sources

arXiv2026

The next-to-top term of the knot Floer homology of some non-fibered knots

Sivek conjectured that the rank of knot Floer homology in the next-to-top Alexander grading is at least the rank in the top Alexander grading. Baldwin and Vela-Vick verified this conjecture in the case of fibered knots arXiv:1801.06563. Ni gave a generalization of this result (for knots in generalized $L$-spaces) to cases in which the knot Floer homology satisfies an algebraic condition arXiv:2104.14687. We give an independent generalization of Baldwin and Vela-Vick's result to a family of knots with Seifert surfaces satisfying certain conditions.

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arXiv2026

Real sutured Heegaard Floer homology

We develop a theory of real sutured manifolds and a real Heegaard Floer theory for these manifolds. We develop a notion of real nice diagrams, and prove that our invariant is combinatorially computable. Our theory shares many structural properties with Juh\'asz's sutured Floer homology, as does the topological theory of real sutured manifolds with Gabai's original sutured manifold theory. We also show that our invariant has several new structural properties differentiating it from sutured Floer homology.

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arXiv2025

On Heegaard Floer minimal knots in sutured manifolds

Li-Xie-Zhang classified instanton Floer minimal knots in balanced sutured manifolds subject to a condition on the fundamental group. In this paper, we give a similar classification in the Heegaard Floer homology setting. Since our classifications agree when they are both applicable, this provides further evidence for the conjecture of Kronheimer-Mrowka that instanton Floer homology and Heegaard Floer homology are isomorphic. We also study link Floer homology botany question in $S^1\times S^2$, showing that link Floer homology detects spherical braid closures among homologically nontrivial links.

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arXiv2025

Degree-1 maps and rank inequalities in Heegaard Floer homology

Ghosh-Sivek-Zentner constructed degree-1 maps from certain rational homology solid tori to the twisted $I$-bundle over the Klein bottle. We show that these maps yield rank inequalities for Heegaard Floer homology. To do so, we use Hanselman-Rasmussen-Watson's immersed curve interpretation of bordered Floer homology, extending their proof of a similar rank inequality corresponding to degree-1 maps to the solid torus. Our result provides further evidence for Kronheimer-Mowka's conjectured relationship between Heegaard Floer homology and instanton Floer homology.

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arXiv2024

Studying links via booklinks: A Markov theorem

We introduce "book links" as a generalization of braids in open book decompositions; this new class of objects includes both braids and plats as special cases. We then prove a version of Markov's theorem in this general setting by extending the theory of open book foliations.

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arXiv2024

Closures of $3$-braids and detection

We give some new link detection results for link Floer homology, Khovanov homology and annular Khovanov homology. The links we detect arise via different closure operations on $3$-braids. Examples of our results include that link Floer homology detects the Mazur link, that annular Khovanov homology detects the Mazur pattern, and that Khovanov homology detects L6a2 and L9n15. The Mazur pattern detection result depends on a new bound on the rank of the annular Khovanov homology of certain links.

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arXiv2024

Floer homology, clasp-braids and detection results

Martin showed that link Floer homology detects braid axes. In this paper we extend this result to give a topological characterisation of links which are almost braided from the point of view of link Floer homology. The result is inspired by work of Baldwin-Sivek and Li-Ye on nearly fibered knots. Applications include that Khovanov homology detects the Whitehead link and $L7n2$, as well as infinite families of detection results for link Floer homology and annular Khovanov homology.

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arXiv2024

Is the geography of Heegaard Floer homology restricted or the $L$-space conjecture false?

In a recent note F. Lin showed that if a rational homology sphere $Y$ admits a taut foliation then the Heegaard Floer module $HF^-(Y)$ contains a copy of $\mathbf{F}[U]/U$ as a summand (arXiv:2309.01222). This implies that either the $L$-space conjecture is false or that Heegaard Floer homology satisfies a geography restriction. We verify that Lin's geography restriction holds for a wide class of rational homology spheres. Indeed, we show that the Heegaard Floer module $HF^-(Y)$ may satisfy a stronger geography restriction.

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arXiv2023

(1,1) almost L-space knots

We give a diagrammatic characterization of the $(1,1)$ knots in the three-sphere and lens spaces which admit large Dehn surgeries to manifolds with Heegaard Floer homology of next-to-minimal rank. This is inspired by a corresponding result for $(1,1)$ knots which admit large Dehn surgeries to manifolds with Heegaard Floer homology of minimal rank due to Greene-Lewallen-Vafaee.

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arXiv2023

The $CFK^\infty$ Type of Almost L-space Knots

Heegaard Floer homology and knot Floer homology are powerful invariants of 3-manifolds and links respectively. L-space knots are knots which admit Dehn surgeries to 3-manifolds with Heegaard Floer homology of minimal rank. In this paper we study almost L-space knots, which are knots admitting large Dehn surgeries to 3-manifolds with Heegaard Floer homology of next-to-minimal rank. Our main result is a classification of the $CFK^\infty$ type of almost L-space knots. As corollaries we show that almost L-space knots satisfy various topological properties, including some given by Baldwin-Sivek. We also give some new cable link detection results.

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arXiv2022

Cable Links, Annuli and Sutured Floer homology

We apply sutured Floer homology techniques to study the knot and link Floer homologies of various links with annuli embedded in their exteriors. Our main results include, for large $m$, characterizations of links with the same link Floer homology as $(n,nm)$-cables of $L$-space knots or the same knot Floer homology as $(2,2m)$-cables of $L$-space knots. Note that Knot Floer homology carries less grading data than Link Floer homology, so the latter characterizations are stronger than former. These characterizations yield some new link detection results.

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arXiv2022

Rank Bounds in Link Floer Homology and Detection Results

Viewing the BRAID invariant as a generator of link Floer homology we generalise work of Baldwin-Vela-Vick to obtain rank bounds on the next to top grading of knot Floer homology. These allow us to classify links with knot Floer homology of rank at most eight, and prove a variant of a classification of links with Khovanov homology of low rank due to Xie-Zhang. In another direction we use a variant of Ozsv\'ath-Szab\'o classification of $E_2$ collapsed $\mathbb{Z} \oplus\mathbb{Z}$ filtered chain complexes to show that knot Floer homology detects $T(2,8)$ and $T(2,10)$. Combining these techniques with the spectral sequences of Batson-Seed, Dowlin, and Lee we can show that Khovanov homology likewise detects $T(2,8)$ and $T(2,10)$.

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arXiv2021

On the nonorientable four-ball genus of torus knots

The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable four-ball genus $\gamma_4$ of any knot. This bound is sharp for several families of torus knots, including $T_{4n,(2n\pm 1)^2}$ for even $n\ge 2$, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever $p$ is an even positive integer and $\frac{p}{2}$ is not a perfect square, the torus knot $T_{p,q}$ does not bound a locally flat M\"obius band for almost all integers $q$ relatively prime to $p$.

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arXiv2020

Knot Floer homology, link Floer homology and link detection

We give new link detection results for knot and link Floer homology inspired by recent work on Khovanov homology. We show that knot Floer homology detects $T(2,4)$, $T(2,6)$, $T(3,3)$, $L7n1$, and the link $T(2,2n)$ with the orientation of one component reversed. We show link Floer homology detects $T(2,2n)$ and $T(n,n)$, for all $n$. Additionally we identify infinitely many pairs of links such that both links in the pair are each detected by link Floer homology but have the same Khovanov homology and knot Floer homology. Finally, we use some of our knot Floer detection results to give topological applications of annular Khovanov homology.

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