A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.
Proves cosmetic crossing conjecture for certain knots.
Proves partial-dual genus polynomial is a knot invariant weight system.
New lower bound for knot genus using Links-Gould invariant.
A link is adequate and has Turaev genus one if its Jones polynomial span is one less than its crossing number.
Paper constructs S-equivalent genus one knots distinguishable by Jones polynomial.
Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 3-variable polynomials, answering a question raised by Dunfield et al in their study of genus 2 mutants. While pairs of genus 2 mutant knots ha…
Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares genus 2 mutation with the better-known Conway mutation in the context of knots in the 3-sphere. Despite the fact that any Conway mutation can …
Paper categorifies a polynomial related to ribbon graphs.
Formula for colored Links-Gould polynomial with genus bounds.
Simple proof of knot genus theorem using Alexander polynomial.
Determines the crossing number of polynomial curve systems on surfaces.
Bounds on knot polynomials for Lie superalgebras of type I.
We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.
Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister t…
We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples…
It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomia…
Study knots with genus one, finds Gordian distance and cosmetic crossing constraints.
Prime knots of genus one admitting diagram with at most five classical crossings were classified by Akimova and Matveev in 2014. In 2018 Kaur, Prabhakar and Vesnin introduced families of L-polynomials and F-polynomials for virtual knots which are generalizations of affine index polynomial. Here we introduce a notion of…
New polynomial connects knot genus to 3-manifold geometry.
The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribb…
Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the…
The Hirzebruch -genus and Poincare polynomial share some similar features. In this article we investigate two of their similar features simultaneously. Through this process we shall derive several new results as well as reprove and improve some known results.
We continue the study of the genus of knot diagrams, deriving a new description of generators using Hirasawa's algorithm. This description leads to good estimates on the maximal number of crossings of generators and allows us to complete their classification for knots of genus 4. As applications of the genus 4 classifi…
Study on frequencies of non-simple curves in surfaces of large genus.
Third coefficient of lens space knots' Alexander polynomials restricts surgeries to specific torus knots.
Paper conjectures Links-Gould invariant generalizes Alexander polynomial.
New bounds on virtual link genus using quantum supergroups.
For an oriented virtual link, L.H. Kauffman defined the f-polynomial (Jones polynomial). The supporting genus of a virtual link diagram is the minimal genus of a surface in which the diagram can be embedded. In this paper we show that the span of the f-polynomial of an alternating virtual link L is determined by the nu…
The paper calculates super Weil-Petersson volumes for large genus.
The paper sets genus bounds for twisted quantum invariants.
New skein theory for Links-Gould polynomial simplifies link evaluations.
Study on knots, genera, and algebraic concordance groups.
Let be a non-split prime alternating link with crossings. We show that for each fixed , the number of genus- Seifert surfaces for is bounded by an explicitly given polynomial in . The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponenti…
Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of …
New knot polynomials reveal patterns and mutations.
In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivia…
Formula for Alexander polynomial of twisted torus knots derived.
We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links b…
Paper analyzes double twist knots using adjoint hyperbolic torsion polynomial.
This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relatio…
We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.
In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus correctio…
The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, co…
The paper calculates ribbon numbers for 12-crossing knots using Alexander polynomials.
We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.
The Turaev genus defines a natural filtration on knots where Turaev genus zero knots are precisely the alternating knots. We show that the signature of a Turaev genus one knot is determined by the number of components in its all-A Kauffman state, the number of positive crossings, and its determinant. We also show that …
We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surf…