Tootfinder

This https://arxiv.org/abs/2502.02939 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On grid homology for diagonal knots
We partially determine grid homology (combinatorial knot Floer homology) of diagonal knots, which are conjectured to be equivalent to positive braid knots, by exploiting nice grid diagrams. We compare diagonal knots to various classes of knots, such as positive braids, fibered positive knots, and $L$-space knots.

Profiles of Critical Flat Ribbon Knots
Jos\'e Ayala
https://arxiv.org/abs/2506.04403 https://arxiv.org/pdf/2506.04403

Profiles of Critical Flat Ribbon Knots
The main open problem in geometric knot theory is to provide a tabulation of knots based on an energy criterion, with the goal of presenting this tabulation in terms of global energy minimisers within isotopy classes, often referred to as ideal knots. Recently, the first examples of minimal length diagrams and their corresponding length values have been determined by Ayala, Kirszenblat, and Rubinstein. This article is motivated by the scarcity of examples despite several decades of intense rese…

This https://arxiv.org/abs/2505.13168 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Knots that share four surgeries
Distinct knots K, K' can sometimes share a common p/q-framed Dehn surgery. A folk conjecture held that for a fixed pair of knots, this can occur for at most one value of p/q. We disprove this conjecture by constructing pairs of distinct knots K,K' that have common Dehn surgeries for four distinct slopes. We also construct non-isotopic Legendrian knots K,K' that have contactomorphic contact (+1)-and (-1)-surgeries, disproving an analogous conjecture in contact geometry.

Smoothly slice knots with smoothly non-approximable topological slice discs
Min Hoon Kim, Mark Powell
https://arxiv.org/abs/2506.03705 https://

Smoothly slice knots with smoothly non-approximable topological slice discs
We construct infinitely many smoothly slice knots having topological slice discs that are non-approximable by smooth slice discs.

This https://arxiv.org/abs/2504.20167 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

Emergence of Hermitian topology from non-Hermitian knots
The non-Hermiticity of the system gives rise to a distinct knot topology in the complex eigenvalue spectrum, which has no counterpart in Hermitian systems. In contrast, the singular values of a non-Hermitian (NH) Hamiltonian are always real by definition, meaning that they can also be interpreted as the eigenvalues of some underlying Hermitian Hamiltonian. In this work, we demonstrate that if the singular values of an NH Hamiltonian are treated as eigenvalues of prototype translational invarian…

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.

A Fox-Neuwirth Basis for the Sinha Spectral Sequence
Andrea Marino
https://arxiv.org/abs/2505.22958 https://arxiv.org/pdf/2505.22958

A Fox-Neuwirth Basis for the Sinha Spectral Sequence
Recently, Sinha defined a spectral sequence approximating the (co)homology of the space of long knots in R^m modulo immersions, stemming from a cosimplicial structure on the compactified configuration spaces à la Kontsevich. We provide an equivalent cosimplicial structure on (the barycentric subdivision of) a regular CW complex with cells indexed by Fox-Neuwirth trees. As a corollary, we give a combinatorial presentation of the Sinha Spectral Sequence in terms of Fox-Neuwirth trees for all dim…

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.

Will we ever decode the writing system used by the Inca?
Gift article, no paywall.
https://www.theatlantic.com/culture/archiv

Unraveling the Secrets of the Inca Empire
For hundreds of years, Andean people recorded information by tying knots into long cords. Will we ever be able to read them?

This https://arxiv.org/abs/2502.10370 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Explicit Formulas for the Alexander Polynomial of Pretzel Knots
We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial.

This https://arxiv.org/abs/2210.04292 has been replaced.
link: https://scholar.google.com/scholar?q=a

Transverse tori in Engel manifolds
We show that tori in Engel 4-manifolds behave analogously to knots in contact 3-manifolds: Every torus with trivial normal bundle is isotopic to infinitely many distinct transverse tori, distinguished locally (and globally in the nullhomologous case) by their formal invariants. (Few examples of transverse tori were previously known.) We classify the formal invariants, which are richer than for transverse knots. We show that in an overtwisted Engel structure, up to homotopy through such structur…

Will we ever decode the writing system used by the Inca?
Gift article, no paywall.
https://www.theatlantic.com/culture/archiv

Unraveling the Secrets of the Inca Empire
For hundreds of years, Andean people recorded information by tying knots into long cords. Will we ever be able to read them?

A generating set of Reidemeister moves of oriented virtual knots
Danish Ali
https://arxiv.org/abs/2505.24094 https://arxiv.org/pdf/25…

A generating set of Reidemeister moves of oriented virtual knots
In oriented knot theory, verifying a quantity is an invariant involves checking its invariance under all oriented Reidemeister moves, a process that can be intricate and time-consuming. A generating set of oriented moves simplifies this by requiring verification for only a minimal subset from which all other moves can be derived. While generating sets for classical oriented Reidemeister moves are well-established, their virtual counterparts are less explored. In this study, we enumerate the ori…

This https://arxiv.org/abs/2409.02192 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A note on cables and the involutive concordance invariants
We prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer ho…

Bordered Heegaard Floer modules for satellite operations using planar graphs
Shikhin Sethi
https://arxiv.org/abs/2506.04222 https://a…

Bordered Heegaard Floer modules for satellite operations using planar graphs
Lipshitz, Ozsváth, and Thurston extend the theory of bordered Heegaard Floer homology to compute $\mathbf{CF}^-$. Like with the hat theory, their minus invariants provide a recipe to compute knot invariants associated to satellite knots. We combinatorially construct the weighted $A_\infty$-modules associated to the $(p, 1)$-cable. The operations on these modules count certain classes of inductively constructed decorated planar graphs. This description of the weighted $A_\infty$-modules provide…

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.

This https://arxiv.org/abs/2406.02957 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A note on the involutive invariants of splices
A natural family of potentially 2-torsion elements in the integer homology cobordism group consists of splices of knots with their mirrors. We show that such 3-manifolds have locally trivial involutive Floer homology. We show some related families of splices also have locally trivial involutive Floer homology. Our arguments show that many gauge theoretic invariants also vanish on these 3-manifolds.

A parallel algorithm for the computation of the Jones polynomial
Kasturi Barkataki, Eleni Panagiotou
https://arxiv.org/abs/2505.23101 https://

A parallel algorithm for the computation of the Jones polynomial
Knots, links and entangled filaments appear in many physical systems of interest in biology and engineering. Classifying knots and measuring entanglement is of interest both for advancing knot theory, as well as for analyzing large data that become available through experiments or Artificial Intelligence. In this context, the efficient computation of topological invariants and other metrics of entanglement becomes an urgent issue. The computation of common measures of topological complexity, su…

Knot invariants from representations of braids by automorphisms of a free group
Vladimir Shpilrain
https://arxiv.org/abs/2505.22403 https://

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.