Tootfinder

The torsion order in knot Floer homology is multiplicative for L-space knots under cabling
David Suchodoll
https://arxiv.org/abs/2506.09577 https://…

The torsion order in knot Floer homology is multiplicative for L-space knots under cabling
The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from the knot Floer homology. It bounds many invariants that are difficult to calculate. For all L-space cables of L-space knots, we prove that $\operatorname{Ord}(K) + 1$ is multiplicative in $p$ when $g(K) > 1$. The same holds for $g(K) = 1$, provided $q > 2p$. We further compute the knot Floer torsion order in the range $p < q < 2p$, thereby determining $\operatorname{Ord}(K_{p,q})$ from $\operatorname{Ord}(K)$ for all …

Closures of 1-tangles and annulus twists
Scott A. Taylor
https://arxiv.org/abs/2506.11226 https://arxiv.org/pdf/2506.11226

Closures of 1-tangles and annulus twists
A 1-tangle is a properly embedded arc $ψ$ in an unknotted solid torus $V$ in $S^3$. Attaching an arc $ϕ$ in the complementary solid torus $W$ to its endpoints creates a knot $K(ϕ)$ called the closure of $ψ$. We show that for a given nontrivial 1-tangle $ψ$ there exist at most two closures that are the unknot. We give a general method for producing nontrivial 1-tangles admitting two distinct closures and show that our construction accounts for all such examples. As an application, we show t…

Knot-Driven Spin Selectivity: Topological Chirality-Induced Robust Spin Polarization in Molecular Knots
Xi Sun, Kai-Yuan Zhang, Shu-Zheng Zhou, Hua-Hua Fu
https://arxiv.org/abs/2507.20172

Knot-Driven Spin Selectivity: Topological Chirality-Induced Robust Spin Polarization in Molecular Knots
Compared to traditional structural chiral materials (e.g., DNA, helicene), topological chirality in trefoil knot molecules has demonstrated multiple remarkable advantages in chirality-induced spin selectivity (CISS), including ultra-high spin polarization of nearly 90%, conductivity increased by two orders of magnitude, and high-temperature stability (up to 350$^{\circ}$C). However, the underlying physical mechanism remains elusive. This work establishes, for the first time, a fundamental theor…

Cubic spline functions revisited
Florian Jarre
https://arxiv.org/abs/2507.05083 https://arxiv.org/pdf/2507.05083

Cubic spline functions revisited
In this paper a fourth order asymptotically optimal error bound for a new cubic interpolating spline function, denoted by Q-spline, is derived for the case that only function values at given points are used but not any derivative information. The bound seems to be stronger than earlier error bounds for cubic spline interpolation in such setting such as the not-a-knot spline. A brief analysis of the conditioning of the end conditions of cubic spline interpolation leads to a modification of the n…

Despite the overwhelming interconnectedness of life's problems, even the smallest actions can send ripples of change through the whole web. full article @ https://www.brichapman.com/p/everything-is-connected-in-a-giant

Everything is Connected (In a Giant Tangled Knot)
welcome to the existential hairball 🐈

Nodal Tangles
Joel Schmitz
https://arxiv.org/abs/2506.23754 https://arxiv.org/pdf/2506.23754

Nodal Tangles
We study piecewise linear knot diagrams in the base of almost toric fibrations of symplectic four-manifolds. These diagrams translate to deformations of the almost toric fibration. We give several applications to symplectic topology, among them a proof of a conjecture by Symington, the calculation of the displacement energy for many fibres of toric moment maps, and an elementary recipe for building and distinguishing Lagrangian torus knots.

This https://arxiv.org/abs/2205.12798 has been replaced.
link: https://scholar.google.com/scholar?q=a

An involutive dual knot surgery formula
We prove an involutive analog of the dual knot surgery formula of Eftekhary and Hedden-Levine. We also compute a small model for the local equivalence class of the involutive dual knot complex.

Cosmetic surgeries on knots in homology spheres and the Casson--Walker invariant
Kazuhiro Ichihara, In Dae Jong
https://arxiv.org/abs/2506.02682 https://…

Cosmetic surgeries on knots in homology spheres and the Casson--Walker invariant
We study cosmetic surgeries on a knot in a homology sphere. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredient is the rational surgery formula of the Casson--Walker invariant for 2-component links in the 3-sphere.

Integer inequalities between knot invariants, skein tree depth and delta-crossing numbers
Michal Jablonowski
https://arxiv.org/abs/2506.00937 https://

Integer inequalities between knot invariants, skein tree depth and delta-crossing numbers
This survey collects known (and one new) inequalities between integer-valued classical knot invariants. It features: a visual graph of the relations; proof of the new inequality relating skein tree depth and delta-crossing numbers; and examples where the inequality becomes equality.

The flip map and involutions on Khovanov homology
Daren Chen, Hongjian Yang
https://arxiv.org/abs/2506.00824 https://arxiv.org/pdf/25…

The flip map and involutions on Khovanov homology
The flip symmetry on knot diagrams induces an involution on Khovanov homology. We prove that this involution is determined by its behavior on unlinks; in particular, it is the identity map when working over $\mathbb{F}_2$. This confirms a folklore conjecture on the triviality of the Viro flip map. As a corollary, we prove that the symmetries on the transvergent and intravergent diagrams of a strongly invertible knot induce the same involution on Khovanov homology. We also apply similar techniqu…

This https://arxiv.org/abs/2505.22403 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Knot invariants from representations of braids by automorphisms of a free group
We describe an alternative way of computing Alexander polynomials of knots/links, based on the Artin representation of the corresponding braids by automorphisms of a free group. Then we apply the same method to other representations of braid groups discovered by Wada and compare the corresponding isotopic invariants to Alexander polynomials.

2-cabling and tangle operators in Khovanov theory
Mihai Marian
https://arxiv.org/abs/2508.00644 https://arxiv.org/pdf/2508.00644

2-cabling and tangle operators in Khovanov theory
We describe an operator on 4-ended tangles that is induced by 2-cabling of a strongly invertible knot. By passing to the 4-ended tangle Khovanov theory of Kotelskiy-Watson-Zibrowius, this induces an operator on the category of type D structures over the Bar-Natan algebra $\mathcal{B}$, as well as on a Fukaya category of the 4-punctured 2-sphere. We provide a full description of this operator's restriction to cap-trivial tangles. Finally, we extract geography results that are inspired by a recen…

Hyperbolic small knots in spherical manifolds
Kazuhiro Ichihara
https://arxiv.org/abs/2506.01041 https://arxiv.org/pdf/2506.01041

Hyperbolic small knots in spherical manifolds
It was conjectured by Lopez that every closed irreducible non-Haken 3-manifold contains a small knot. In this paper, we give explicit examples of hyperbolic small knots in most closed orientable spherical 3-manifolds other than prism manifolds.

Replaced article(s) found for math.GT. https://arxiv.org/list/math.GT/new
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- Trace-free characters and abelian knot contact homology I
Fumikazu Nagasato