Tootfinder

Cyclic $\mathcal{U}_\xi(\mathfrak{sl}_2)$-modules and invariants of knots with flat $\mathfrak{sl}_2$ connections in the complement
Calvin McPhail-Snyder, Nicolai Reshetikhin
https://arxiv.org/abs/2509.02354

Cyclic $\mathcal{U}_ξ(\mathfrak{sl}_2)$-modules and invariants of knots with flat $\mathfrak{sl}_2$ connections in the complement
The main result of this paper is the factorization of the holonomy $R$-matrix for quantum $\mathfrak{sl}_2$ at a root of unity into a product of four quantum holonomy dilogarithms. This factorization extends previously known results in this direction. We collect many existing results needed for the factorization. We use the holonomy $R$-matrices to define representations of a groupoid of braids with flat $\mathfrak{sl}_{2}$ connections, which also define invariants of knots with such connection…

Constructing and Cataloging 2-Adjacent Knots
John Carney, Everett Meike
https://arxiv.org/abs/2510.00291 https://arxiv.org/pdf/2510.00291

Constructing and Cataloging 2-Adjacent Knots
Generalizing unknotting number, $n$-adjacent knots have $n$ crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer $d$-invariants and the Alexander polynomial, we develop a new technique to obstruct 2-adjacency, and we prove conjectures of Ito and Kato regarding 2-adjacent knots.

Maps from knots to $2$-links, chord diagrams, and a way to enhance Vassiliev invariants
Vassily Olegovich Manturov
https://arxiv.org/abs/2509.01971 https://

Maps from knots to $2$-links, chord diagrams, and a way to enhance Vassiliev invariants
In the present paper, we discuss a way of generalising Vassiliev knot invariants and weight systems to framed chord diagrams having framing 0 and 1.

Affine representations of rational and pretzel knots
Javier Mart\'inez
https://arxiv.org/abs/2510.02077 https://arxiv.org/pdf/2510.02077

Affine representations of rational and pretzel knots
We construct and study representations of rational and pretzel tangle and knot groups into the affine group $\mathrm{AGL}(1,\mathbb{C})$, via a TQFT that is valued in the category of spans of singular vector bundles over $\mathbb{C}^{\ast}$. For these families, we derive closed-form expressions for their Alexander polynomials and establish bounds on their zeros. Finally, we specialize the functor at $t=-1$ and analyze colorings of rational tangles in terms of spans of complex vector spaces.

Knots and variance ordering of sequential Monte Carlo algorithms
Joshua J Bon, Anthony Lee
https://arxiv.org/abs/2510.01901 https://arxiv.org/pdf/2510.0190…

Knots and variance ordering of sequential Monte Carlo algorithms
Sequential Monte Carlo algorithms, or particle filters, are widely used for approximating intractable integrals, particularly those arising in Bayesian inference and state-space models. We introduce a new variance reduction technique, the knot operator, which improves the efficiency of particle filters by incorporating potential function information into part, or all, of a transition kernel. The knot operator induces a partial ordering of Feynman-Kac models that implies an order on the asymptot…

Structure groups and second homology groups of linear Alexander quandles
Adrien Cl\'ement
https://arxiv.org/abs/2510.00830 https://arxiv.org/pdf/2510.0…

Structure groups and second homology groups of linear Alexander quandles
Quandles are self-distributive algebraic structures known as sources of strong knots invariants, but also appearing in other contexts. From any quandle, one can construct two invariants: the structure group and the second quandle homology group. These groups are useful in applications, but hard to compute. In this paper, we focus on Alexander quandles over a cyclic group $\mathbb{Z}_n$. By using explicit rewriting techniques, we show that the structure group of such a quandle injects into $\mat…

Exact integral formulas for volumes of two-bridge knot cone-manifolds
Anh T. Tran, Nisha Yadav
https://arxiv.org/abs/2510.02095 https://arxiv.org/pdf/2510.…

Exact integral formulas for volumes of two-bridge knot cone-manifolds
We provide exact integral formulas for hyperbolic and spherical volumes of cone-manifolds whose underlying space is the $3$-sphere and whose singular set belongs to three infinite families of two-bridge knots: $C(2n,2)$ (twist knots), $C(2n,3)$, and $C(2n,-2n)$ for any non-zero integer $n$. Our formulas express volumes as integrals of explicit rational functions involving Chebyshev polynomials of the second kind, with integration limits determined by roots of algebraic equations. This extends p…

Crosslisted article(s) found for math.QA. https://arxiv.org/list/math.QA/new
[1/1]:
- Affine representations of rational and pretzel knots
Javier Mart\'inez

Extrinsic systole of Seifert surfaces and distortion of knots
Sahana Vasudevan
https://arxiv.org/abs/2510.01095 https://arxiv.org/pdf/2510.01095

Extrinsic systole of Seifert surfaces and distortion of knots
In 1983, Gromov introduced the notion of distortion of a knot, and asked if there are knots with arbitrarily large distortion. In 2011, Pardon proved that the distortion of $T_{p,q}$ is at least $\min\{p,q\}$ up to a constant factor. We prove that the distortion of $T_{p, p+1}\# K$ is at least $p$ up to a constant, independent of $K$. We also prove that any embedding of a minimal genus Seifert surface for $T_{p,p+1}\# K$ in $\mathbb{R}^3$ has small extrinsic systole, in the sense that it contai…

Full twists and stability of knots and quivers
Sachin Chauhan, Piotr Kucharski, Dmitry Noshchenko, Ramadevi Pichai, Vivek Kumar Singh, Marko Sto\v{s}i\'c
https://arxiv.org/abs/2508.18417

Full twists and stability of knots and quivers
We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a~full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmente…

Embedding calculus and Vassiliev spectral sequence
Syunji Moriya
https://arxiv.org/abs/2509.23766 https://arxiv.org/pdf/2509.23766

Embedding calculus and Vassiliev spectral sequence
Vassiliev spectral sequence and Sinha spectral sequence are both related to cohomology of the space of long knots $\mathbb{R}\to \mathbb{R}^3$. Although they have different origins, the Vassiliev $E_1$-page and the Sinha $E_2$-page are isomorphic (up to a degree shift). In this paper, we prove that they have isomorphic $E_\infty$-pages if the coefficient ring is a field. Together with degeneracy of the Sinha sequence, this implies that the Vassiliev sequence degenerates at $E_1$-page over $\mat…

Non-regular Lagrangian concordances between Lagrangian fillable Legendrian knots
Georgios Dimitroglou Rizell, Roman Golovko
https://arxiv.org/abs/2509.13594 https://

Non-regular Lagrangian concordances between Lagrangian fillable Legendrian knots
In this short note, we construct a family of non-regular, and therefore non-decomposable, Lagrangian concordances between Lagrangian fillable Legendrian knots in the standard contact 3-dimensional sphere. More precisely, for every decomposable Lagrangian concordance from $Λ_-$ to $Λ_+$, where $Λ_{\pm}$ are smoothly non-isotopic Legendrian knots, we construct a non-regular Lagrangian concordance from $Λ'_+$ to $Λ'_-$, where both $Λ'_\pm$ are Lagrangian fillable Legendrian knots obtained fr…

Ribbon concordance and fibered predecessors
John A. Baldwin, Steven Sivek
https://arxiv.org/abs/2510.02214 https://arxiv.org/pdf/2510.02214

Ribbon concordance and fibered predecessors
Given any knot K in the 3-sphere, we prove that there are only finitely many hyperbolic fibered knots which are ribbon concordant to K. It follows that every fibered knot in the 3-sphere has only finitely many hyperbolic predecessors under ribbon concordance. Our proof combines results about maps on Floer homology induced by ribbon cobordisms with a relationship between the knot Floer homology of a fibered knot and fixed points of its monodromy. We then use the same techniques in combination wi…

Fusion and Fission of Particle-like Chiral Nematic Vortex Knots
Darian Hall, Jung-Shen Benny Tai, Louis H. Kauffman, Ivan I. Smalyukh
https://arxiv.org/abs/2508.05841 https://…

Fusion and Fission of Particle-like Chiral Nematic Vortex Knots
Vortex knots have been seen decaying in many physical systems. Here we describe topologically protected vortex knots, which remain stable and undergo fusion and fission while conserving a topological invariant analogous to that of baryon number. While the host medium, a chiral nematic liquid crystal, exhibits intrinsic chirality, cores of the vortex lines are structurally achiral regions where twist cannot be defined. We refer to them as "dischiralation" vortex lines, in analogy to dislocations…

On Steenrod squares for even and odd Khovanov homology
Advika Rajapakse
https://arxiv.org/abs/2509.03396 https://arxiv.org/pdf/2509.03396

On Steenrod squares for even and odd Khovanov homology
For an arbitrary link $L \subset S^3$ , Sarkar-Scaduto-Stoffregen construct a family of spatial refinements of even and odd Khovanov homology. We give a computation of $\text{Sq}^2$ on these spaces, determining their stable homotopy types for all knots K up to 11 crossings. We also prove that the Steenrod squares $\text{Sq}_0^2$ , $\text{Sq}_1^2$ defined by Schütz do arise as Steenrod squares on these spaces.

Replaced article(s) found for math.AT. https://arxiv.org/list/math.AT/new
[1/1]:
- Sinha's spectral sequence for long knots in codimension one and non-formality of the little 2-dis...
Syunji Moriya

$h$-Principles for smooth curves and knots with prescribed curvature
Mohammad Ghomi, Matteo Raffaelli
https://arxiv.org/abs/2510.05275 https://arxiv.org/pd…

$h$-Principles for smooth curves and knots with prescribed curvature
We show that smooth curves with prescribed curvature satisfy a $C^1$-dense $h$-principle in the space of immersed curves in Euclidean space. More precisely, every $C^{α\geq 2}$ curve with nonvanishing curvature in $R^{n\geq 3}$ can be $C^1$-approximated by $C^α$ curves of any larger curvature, prescribed as a function of arclength. It follows that there exist $C^\infty$ knots of prescribed curvature in every isotopy class of closed curves embedded in $R^3$.

The knots people twist themselves into to defend Perplexity is super funny.
Motivated reasoning is one hell of a drug.

Statistics in 3d gravity from knots and links
Jeevan Chandra
https://arxiv.org/abs/2508.10864 https://arxiv.org/pdf/2508.10864

Statistics in 3d gravity from knots and links
In recent years, there has been remarkable progress in evaluating wormhole amplitudes in 3d Einstein gravity with negative cosmological constant and matching them to statistics of 2d CFT data. In this work, we compute non-perturbative Gaussian and non-Gaussian gravitational contributions to the OPE statistics using a framework that can systematically generate a class of such non-perturbative effects - \textit{Fragmentation of knots and links by Wilson lines}. We illustrate this idea by construc…

The first proper motion measurement of the acceleration regions in the large-scale jets of SS 433 powering the W50 nebula
Naomi Tsuji, Yoshiyuki Inoue, Dmitry Khangulyan, Kaya Mori, Samar Safi-Harb, Takaaki Tanaka, Laura Olivera-Nieto, Brydyn Mac Intyre, Kazuho Kayama, Takeshi Go Tsuru, Hiroyuki Uchida, Tatsuki Fujiwara, Felix Aharonian
https://

The first proper motion measurement of the acceleration regions in the large-scale jets of SS 433 powering the W50 nebula
We report on new Chandra ACIS-I observations of the X-ray knots located in the western and eastern lobes of W50 associated with the parsec-scale jets of the Galactic microquasar SS 433. These knots are likely counterparts of the recently detected very-high-energy ($E>100$ GeV) gamma-ray emission by HAWC and H.E.S.S. These findings, together with the ultra-high-energy signal recently reported by the LHAASO collaboration, have established the SS 433/W50 system as a unique jet-driven PeVatron cand…

Slice depth of 2-knots
Ayaka Ise
https://arxiv.org/abs/2509.16905 https://arxiv.org/pdf/2509.16905

Slice depth of 2-knots
We introduce the notion of slice depth of a 2-knot K, which is the minimal integer n such that K is n-slice. We give an upper bound for the slice depth of the n-twist spin of a classical knot which belongs to several specific classes, namely, certain types of 2-bridge knots, pretzel knots, ribbon knots of 1-fusion, and knots with given unknotting number.

Eagles vs. Broncos takeaways: Bo Nix, Denver hand Philadelphia its first loss https://www.nytimes.com/athletic/6691825/2025/10/05/broncos-eagles-score-result-takeaways-nfl/

Knot quandles distinguish Suciu's ribbon knots
Jumpei Yasuda
https://arxiv.org/abs/2508.15129 https://arxiv.org/pdf/2508.15129

Knot quandles distinguish Suciu's ribbon knots
The knot quandle is an invariant of $n$-knots. In this note, we study the knot quandles of Suciu's ribbon $n$-knots, an infinite family of knots with isomorphic knot groups. We prove that their knot quandles are mutually non-isomorphic. Furthermore, we compute types of these quandles.

Almost-concordance of knots in aspherical 3-manifolds
Ryan Stees
https://arxiv.org/abs/2508.14638 https://arxiv.org/pdf/2508.14638

Almost-concordance of knots in aspherical 3-manifolds
In this paper, we study topological concordance modulo local knotting, or almost-concordance, of knots in 3-manifolds $M\neq S^3$. A. Levine, Celoria (arXiv:1602.05476v4), and Friedl-Nagel-Orson-Powell (arXiv:1611.09114v2) conjecture that, absent the presence of an embedded dual 2-sphere, any free homotopy class $x$ of knots in $M$ contains infinitely many concordance classes modulo the action of the concordance group of knots in $S^3$ by local knotting. We develop a method for confirming this …

Replaced article(s) found for math.DS. https://arxiv.org/list/math.DS/new
[1/1]:
- A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots
Timur Medvedev, Olga Pochinka

On the colored Links--Gould polynomial
Stavros Garoufalidis, Matthew Harper, Rinat Kashaev, Ben-Michael Kohli, Emmanuel Wagner
https://arxiv.org/abs/2509.10911 https://

On the colored Links--Gould polynomial
We give a cabling formula for the Links--Gould polynomial of knots colored with a $4n$-dimensional irreducible representation of $\mathrm{U}^H_q\mathfrak{sl}(2|1)$ and identify them with the $V_n$-polynomial of knots for $n=2$. Using the cabling formula, we obtain genus bounds and a specialization to the Alexander polynomial for the colored Links--Gould polynomial that is independent of $n$, which implies corresponding properties of the $V_n$-polynomial for $n=2$ conjectured in previous wo…

JWST observations of the Ring Nebula (NGC 6720) -- II. PAH emission
Nicholas Clark, Els Peeters, Nick L. J. Cox, Jan Cami, Michael J. Barlow, Patrick J. Kavanagh, Griet Van de Steene, Mikako Matsuura, Albert Zijlstra, Isabel Aleman, Jeronimo Bernard-Salas, Harriet L. Dinerstein, Kay Justtanont, Kyle F. Kaplan, Arturo Manchado, Raghvendra Sahai, Peter van Hoof, Kevin Volk, Roger Wesson

JWST observations of the Ring Nebula (NGC 6720) -- II. PAH emission
Polycyclic aromatic hydrocarbons (PAHs) and carbonaceous dust have been observed in clumpy circumstellar environments, yet their formation and evolutionary pathways in such environments remain elusive. We aim to characterize the PAH emission in a clumpy planetary nebula to decipher their formation and evolution pathways. We obtained JWST Near-Infrared Spectrograph (NIRSpec) and Mid-Infrared Instrument (MIRI) integral field unit spectroscopic observations of two individual knots in the Ring Nebu…

Unique Surgery Descriptions along Knots
Marc Kegel, Misha Schmalian
https://arxiv.org/abs/2508.18521 https://arxiv.org/pdf/2508.18521

Unique Surgery Descriptions along Knots
We prove that for any non-trivial knot K, infinitely many r-surgeries K(r) along K have a unique surgery description along a knot. Moreover, we show that for any hyperbolic L-space knot K and infinitely many integer slopes n, the manifold K(n) has a unique surgery description. Here we say a 3-manifold M has a unique surgery description along a knot in S^3 if there is a unique pair (K,r) of a knot K and a slope r such that M is orientation-preservingly diffeomorphic to K(r). This generalises the…

Knotted DNA Configurations in Bacteriophage Capsids: A Liquid Crystal Theory Approach
Pei Liu, Zhijie Wang, Tamara Christiani, Mariel Vazquez, M. Carme Calderer, Javier Arsuaga
https://arxiv.org/abs/2509.10410

Knotted DNA Configurations in Bacteriophage Capsids: A Liquid Crystal Theory Approach
Bacteriophages, viruses that infect bacteria, store their micron long DNA inside an icosahedral capsid with a typical diameter of 40 nm to 100 nm. Consistent with experimental observations, such confinement conditions induce an arrangement of DNA that corresponds to a hexagonal chromonic liquid-crystalline phase, and increase the topological complexity of the genome in the form of knots. A mathematical model that implements a chromonic liquid-crystalline phase and that captures the changes in t…

Satellite Operations and $\theta$
Rob McConkey, Luke J Seaton
https://arxiv.org/abs/2509.22939 https://arxiv.org/pdf/2509.22939

Satellite Operations and $θ$
We study the behavior of the knot invariant $θ$ under satellite operations. First, we prove that $θ$ is additive under connected sum. We then introduce a computational tool to generate $t$-twisted Whitehead doubles and apply it to explore the case of untwisted Whitehead doubles. We propose a conjecture describing the behavior of $θ$ on untwisted Whitehead doubles and verify the conjecture for the first 2977 prime knots. The pair of invariants $Θ= (Δ,θ)$ was introduced by Bar-Natan and van…

Inverted Habiro Series and its Residues
Josef Svoboda
https://arxiv.org/abs/2509.22610 https://arxiv.org/pdf/2509.22610

Inverted Habiro Series and its Residues
We study the Gukov--Manolescu (GM) series of knots and the inverted Habiro series (IHS) proposed by S. Park. We give a new formula for IHS in terms of coefficients of the GM series and truncated theta functions. We prove a multiplication formula for IHS, constructing a natural ring, in analogy with work of Habiro. We study the residues of IHS and apply them to Dehn surgery formulas. We also give a curious relation between the asymptotics of the GM series at roots of unity and the Kashaev invari…

Persistent foliarity of (1,1) almost L-space knots
Qingfeng Lyu
https://arxiv.org/abs/2510.11533 https://arxiv.org/pdf/2510.11533

Persistent foliarity of (1,1) almost L-space knots
We propose a branched surface model that could be used to prove (1,1) non-L-space knots in $S^3$ and lens spaces are persistently foliar. We verify that this model works for (1,1) almost L-space knots.

🇺🇦 #NowPlaying on BBCRadio3's #RoundMidnight
Ibelisse Guardia Ferragutti & Frank Rosaly:
🎵 WRITING WITH KNOTS
#IbelisseGuardiaFerragutti #FrankRosaly
https://open.spotify.com/track/1lsaOiMiSgLNovLDTj73mP

On asymmetric hyperbolic L-space knots of braid index four
Kenneth L. Baker, Masakazu Teragaito
https://arxiv.org/abs/2510.07618 https://arxiv.org/pdf/2510…

On asymmetric hyperbolic L-space knots of braid index four
A knot is called an L-space knot if it admits a positive Dehn surgery yielding an L-space. In the SnapPy census, there are exactly 9 asymmetric L-space knots. Among them, the knot t12533 is the only known example of braid index 4. We generalize this knot, and give the first infinite family of asymmetric hyperbolic L-space knots of braid index 4.

Fibering of double twist knots via the adjoint hyperbolic torsion polynomial
Anh T. Tran
https://arxiv.org/abs/2508.14731 https://arxiv.org/pdf/2508.14731

Fibering of double twist knots via the adjoint hyperbolic torsion polynomial
For a hyperbolic knot $K$ in $S^3$, the adjoint hyperbolic torsion polynomial $\mathcal T^{\mathrm{Ad}}_K(t) \in \mathbb C[t^{\pm 1}]$ is defined as a normalization of the twisted Alexander polynomial of $K$ associated with the $\mathrm{SL}_3(\mathbb C)$-representation obtained by composing the holonomy representation of $K$ with the adjoint action of $\mathrm{SL}_2(\mathbb C)$ on its Lie algebra $\mathfrak{sl}_2(\mathbb C)$. In this paper we consider the adjoint hyperbolic torsion polynomial f…

$SU(2)$-representations of Branched Covers
Sudipta Ghosh, Zhenkun Li, Juanita Pinz\'on-Caicedo
https://arxiv.org/abs/2508.19669 https://arxiv.org/pdf/2…

$SU(2)$-representations of Branched Covers
We study the existence of irreducible $SU(2)$-representations for cyclic branched covers of knots in $S^3$. Our main result establishes that if $K$ is a non-trivial prime knot and $d$ is an integer such that $d \geq 2$ and $Σ_d(K)$ is an integer homology sphere, then $π_1(Σ_d(K))$ admits an irreducible $SU(2)$-representation, whenever $K$ satisfies one of two conditions: either $K$ is $2$-periodic, or $K$ can be represented as the closure of a tangle adapted to a $d\times d$ SICUP matrix. Th…

Quantum equations for knots
Thomas Fiedler, Butian Zhang
https://arxiv.org/abs/2509.08423 https://arxiv.org/pdf/2509.08423…

Quantum equations for knots
This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion. If a string link is the product of some string link diagram $T$ and the parallel n-cable of a framed long knot diagram $D$, then there is a canonical arc $push$ in $M_n$, defined by pushing $T$ through the n-cable of $D$. In this paper we apply the combinat…

Crosslisted article(s) found for math.QA. https://arxiv.org/list/math.QA/new
[1/1]:
- Quantum Cocycle Invariants of Knots from Yang-Baxter Cohomology
Masahico Saito, Emanuele Zappala

Khovanov homology of tangles: algorithm and computation
Li Shen, Jian Liu, Guo-Wei Wei
https://arxiv.org/abs/2508.14404 https://arxiv.org/pdf/2508.14404

Khovanov homology of tangles: algorithm and computation
Knot, link, and tangle theory is crucial in both mathematical theory and practical application, including quantum physics, molecular biology, and structural chemistry. Unlike knots and links, tangles impose more relaxed constraints, allowing the presence of arcs, which makes them particularly valuable for broader applications. Although Khovanov homology for knots and links has been extensively studied, its computation for tangles remains largely unexplored. In our recent work, we provide a topo…

The knot quandles of Suciu's ribbon $n$-knots and automorphisms on the free group of rank two
Takuya Sakasai, Kokoro Tanaka
https://arxiv.org/abs/2509.07395 https://

The knot quandles of Suciu's ribbon $n$-knots and automorphisms on the free group of rank two
Yasuda recently proved that the knot quandles of Suciu's $n$-knots, which share isomorphic knot groups, are mutually non-isomorphic. In this paper, we provide an alternative proof of this result by analyzing the conjugacy classes of certain automorphisms of the free group of rank two.

Every negative amphichiral knot is rationally slice
Alessio Di Prisa, Jaewon Lee, O\u{g}uz \c{S}avk
https://arxiv.org/abs/2509.21140 https://arxiv.org/pdf/…

Every negative amphichiral knot is rationally slice
In 2009, Kawauchi proved that every strongly negative amphichiral knot is rationally slice. However, as shown by Hartley in 1980, there are examples of negative amphichiral knots that are not strongly negative amphichiral. In this paper, we prove that every negative amphichiral link whose amphichiral map preserves each component is rationally slice. Our proof relies on a systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior. More…

A stud of 2-periodic weft-knitted textiles using the theory of knots and links
Miriam Kuzbary, Shashank G. Markande, Elisabetta A. Matsumoto, Stanley Pritchard
https://arxiv.org/abs/2510.08384

A stud of 2-periodic weft-knitted textiles using the theory of knots and links
In this study, we use a correspondence between two-periodic weft-knitted textiles and links in the thickened torus to study the former using link invariants. We establish a criterion to identify the set of links whose elements are realized through techniques of weft-knitting leading to new, unconventional types of weft-knitting stitch patterns. A crucial topological underpinning of these links is shown to be their correspondence with ribbon knots and links in Euclidean three-space and equivalen…

Crosslisted article(s) found for math.GT. https://arxiv.org/list/math.GT/new
[1/1]:
- Full twists and stability of knots and quivers
Chauhan, Kucharski, Noshchenko, Pichai, Singh, Sto\v{s}i\'c

Knot primality: knot Floer homology, metacyclic representations and twisted homology
Samantha Allen, Charles Livingston
https://arxiv.org/abs/2508.08102 https://

Knot primality: knot Floer homology, metacyclic representations and twisted homology
We develop purely algebraic methods for proving that a knot is prime. Our approach uses the Heegaard Floer polynomial in conjunction with classical knot-theoretic methods: cyclic, dihedral, and metacyclic covering spaces. The theory of twisted homology allows us to view these approaches from a unified perspective. Collectively, the primality tests developed here have proved primality for over 99.6% of knots in large families of prime knots, including all prime knots with 15 or fewer crossings.

Quasi-alternating surgeries on asymmetric L-space knots in the census
Masakazu Teragaito
https://arxiv.org/abs/2510.04403 https://arxiv.org/pdf/2510.04403

Quasi-alternating surgeries on asymmetric L-space knots in the census
In the SnapPy census, there are 9 asymmetric L-space knots. It is known that each of them admits exactly two quasi-alternating surgeries with the aid of a computer. The purpose of this article is to confirm these surgeries by the Montesinos trick.

On the arc index and Turaev genus of a link
\'Alvaro Del Valle V\'ilchez, Adam M. Lowrance
https://arxiv.org/abs/2509.21274 https://arxiv.org/pdf/2…

On the arc index and Turaev genus of a link
We compute the arc index of an adequate link and establish bounds on the arc index of the closure of a positive 3-braid. We also conjecture an inequality between the crossing number, arc index, and Turaev genus of a link and show the conjecture is true for several infinite families of links including alternating links, links with Turaev genus one, adequate links, closures of positive 3-braids, torus links, and most Kanenobu knots.

Non-Simple knots in Contact 3-Manifolds
Ipsita Datta, Tanushree Shah
https://arxiv.org/abs/2510.02932 https://arxiv.org/pdf/2510.02932

Non-Simple knots in Contact 3-Manifolds
We present new families of examples of non-simple prime Legendrian and transversal knots in tight Lens spaces, which demonstrate that the botany of Legendrians in Lens space is rich. In fact, there are more non-isotopic Legendrians that are topologically isotopic to the $n$-twist knot in a Lens space $L(α, β)$ than in $S^3$. We also include connect sum formulas for rational variants of classical invariants, $\mathrm{tb}_\mathbb{Q}$, $\mathrm{rot}_\mathbb{Q}$, and $\mathrm{sl}_\mathbb{Q}$, whi…

Quantum Cocycle Invariants of Knots from Yang-Baxter Cohomology
Masahico Saito, Emanuele Zappala
https://arxiv.org/abs/2509.04267 https://arxiv.org/pdf/250…

Quantum Cocycle Invariants of Knots from Yang-Baxter Cohomology
Yang-Baxter operators (YBOs) have been employed to construct quantum knot invariants. More recently, cohomology theories for YBOs have been independently developed, drawing inspiration from analogous theories for quandles and other discrete algebraic structures. Quandle cohomology, in particular, gives rise to cocycle invariants for knots via $2$-cocycles, which are closely related to quandle extensions, and knotted surfaces via $3$-cocycles. These quandle cocycle knot invariants have also been…

Applications of the L-space satellite formula
Daren Chen, Ian Zemke, Hugo Zhou
https://arxiv.org/abs/2509.20288 https://arxiv.org/pdf/2509.20288

Applications of the L-space satellite formula
We give a formula for the $τ$-invariant of a satellite knot $P(K,n)$ when $P$ is an L-space satellite operator. Our formula holds for general L-space satellite operators $P$ when the companion $K$ satisfies $ε(K)=1$. When $ε(K)$ is $0$ or $-1$, we state a formula which requires some additional assumptions on $P$ or $n$. Our main tool is our algorithm which computes the knot Floer complex of satellite knots constructed using L-space satellite operators, which we developed in a previous paper.…

Transverse fibered knots and Taut Foliations
Rithwik Susheel Vidyarthi
https://arxiv.org/abs/2509.04680 https://arxiv.org/pdf/2509.04680

Transverse fibered knots and Taut Foliations
We give a short alternative proof of Honda-Kazec-Matic's result, which states that a fibered knot with pseudo-Anosov monodromy and fractional Dehn twist coefficient $\geq1$ supports a contact structure that is a perturbation of a taut foliation.

Surgeries on knots and tight contact structures
Zhenkun Li, Shunyu Wan, Hugo Zhou
https://arxiv.org/abs/2510.05294 https://arxiv.org/pdf/2510.05294

Surgeries on knots and tight contact structures
For any knot $K$ in $S^3$ and any positive rational $r$, we show that smooth $(-r)$-surgery on $K$ always admits a tight contact structure. More specifically, the tightness is detected by the non-vanishing Heegaard Floer contact invariant.

Linear Bound for Kakimizu Complex Diameter of Hyperbolic Knots
Xiao Chen, Wujie Shen
https://arxiv.org/abs/2508.03353 https://arxiv.org/pdf/2508.03353

Linear Bound for Kakimizu Complex Diameter of Hyperbolic Knots
We show that the upper bound of the diameter of the Kakimizu complex of an atoroidal knot grows linearly with the knot genus $g$. Specifically, the diameter is at most $2$ when $g = 1$, at most $6$ when $g = 2$, and at most $4g - 3$ for $g \geq 3$. This confirms a conjecture of Sakuma--Shackleton.

Dehn surgery functions are never injective
Kyle Hayden, Lisa Piccirillo, Laura Wakelin
https://arxiv.org/abs/2508.13369 https://arxiv.org/pdf/2508.13369

Dehn surgery functions are never injective
We prove that, for each fixed rational number $p/q \in \mathbb{Q}$, there exists a pair of distinct knots whose $p/q$-surgeries are orientation-preservingly homeomorphic. This confirms a 1978 conjecture of Gordon.

Psyquandle Brackets
Sam Nelson, Natsumi Oyamaguchi
https://arxiv.org/abs/2508.13331 https://arxiv.org/pdf/2508.13331

Psyquandle Brackets
We extend the notion of biquandle brackets to the case of psyquandles, defining quantum enhancements of the psyquandle counting invariant for singular knots and pseudoknots. We provide examples to illustrate the computation of these invariants, establishing that the enhancement is proper. We compute a few toy examples, noting that the true power of this infinite family of invariants lies in more computationally expensive larger-cardinality psyquandles and infinite coefficient rings.

Replaced article(s) found for math.GT. https://arxiv.org/list/math.GT/new
[1/1]:
- Cable knots are not thin
Subhankar Dey
https://arxiv.or…

Rigidity of highly twisted plat diagrams
Nir Lazarovich, Yoav Moriah, Tali Pinsky, Jessica S. Purcell
https://arxiv.org/abs/2508.07303 https://arxiv.org/pd…

Rigidity of highly twisted plat diagrams
In this paper we prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a $2m$-plat projection, where $m$ is at least four, and height at least two, and each twist region of the plat contains at least four crossings, then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these lin…

Non-characterizing slopes for more 3-manifolds
Matthew Elpers
https://arxiv.org/abs/2510.05412 https://arxiv.org/pdf/2510.05412

Non-characterizing slopes for more 3-manifolds
For any homotopy class h in any compact orientable 3-manifold M which is closed or has exclusively torus boundary components, we produce infinitely many pairs of distinct knots representing h with orientation-preserving homeomorphic 0-surgeries.

Crosslisted article(s) found for math.GT. https://arxiv.org/list/math.GT/new
[1/1]:
- $h$-Principles for smooth curves and knots with prescribed curvature
Mohammad Ghomi, Matteo Raffaelli

A note on real Heegaard Floer homology and localization
Kristen Hendricks
https://arxiv.org/abs/2508.03897 https://arxiv.org/pdf/2508.03897

A note on real Heegaard Floer homology and localization
We prove the existence of a localization spectral sequence for the hat variant of Guth and Manolescu's recent construction of real Heegaard Floer homology, and apply it to branched double covers and strongly invertible knots. Our construction applies to real Lagrangian Floer homology in exact symplectic manifolds equipped with anti-symplectic involutions more generally, and may be of independent interest to symplectic geometers.

Trisected Rainbows and Braids
Rom\'an Aranda, Scott Carter, Julia Courtney, Puttipong Pongtanapaisan
https://arxiv.org/abs/2510.04248 https://arxiv.org…

Trisected Rainbows and Braids
New explicit procedures for passing among triplane diagrams, braid movies, and braid charts for knotted surfaces in $\mathbb{R}^4$ are presented. To this end, rainbow diagrams, which lie between braid charts and triplanes, are introduced. Inequalities relating the braid index and the bridge index of 2-knots are obtained via these procedures. Another consequence is a 4-dimensional version of the classical result that ``the minimal number of Seifert circles equals the braid index of a link'' due …

Replaced article(s) found for math.GT. https://arxiv.org/list/math.GT/new
[1/1]:
- Milnor's invariants for knots and links in closed orientable 3-manifolds
Ryan Stees