Polynomial representation for long knots

Polynomials of genus one prime knots of complexity at most five

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 totally flat-trivial knots and demonstrate that for such knots F-polynomials and L-polynomials coincide with affine index polynomial. We prove that all Akimova - Matveev knots are totally flat-trivial and calculate their affine index polynomials.

A polynomial time knot polynomial

We present the strongest known knot invariant that can be computed effectively (in polynomial time).

Numerical semigroups, Dedekind sums and Alexander polynomials of torus knots

We connect Dedekind sums and Alexander polynomials of torus knots.

Twisted Alexander polynomials of 2-bridge knots

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

Alexander polynomials of doubly primitive knots

We give a formula for Alexander polynomials of doubly primitive knots.

Nonalternating knots and Jones polynomials

Although most knots are nonalternating, modern research in knot theory seems to focus on alternating knots. We consider here nonalternating knots and their properties. Specifically, we show certain classes of knots have nontrivial Jones polynomials.

Cobordism of algebraic knots defined by Brieskorn polynomials

In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.

On the HOMFLY polynomial of 4-plat presentations of knots

In this paper, I give a method to calculate the HOMFLY polynomials of knots by using a representation of the braid group B4 into a group of 3 ? 3 matrices. Also, I will give examples of a 2-bridge knot and a 3-bridge knot that have the same Jone polynomial, but different HOMFLY polynomials.

On Jones polynomials of alternating pretzel knots

We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.

Virtual knots undetected by 1 and 2-strand bracket polynomials

Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, $ K_D $ and $ K_m $, that are not not detected by the bracket polynomial or the 2-strand bracket polynomial. From these diagrams, we construct two infinite families of non-classical virtual knot diagrams that are not d…

The warping polynomial of a knot diagram

We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, and we give an inequality between the span and the dealternating number.

A note on Riley polynomials of $2$-bridge knots

In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.

A characterization of the Murasugi polynomial of an equivariant slice knot

We characterize the Murasugi polynomial of an equivariant slice knot by proving a conjecture of J. Davis and S. Naik.

An explicit formula for the A-polynomial of twist knots

We extend Hoste-Shanahan's calculations for the A-polynomial of twist knots, to give an explicit formula.

On the Kawauchi conjecture about the Conway polynomial of achiral knots

We give a counterexample to the Kawauchi conjecture on the Conway polynomial of achiral knots which asserts that the Conway polynomial $C(z)$ of an achiral knot satisfies the splitting property $C(z)=F(z)F(-z)$ for a polynomial $F(z)$ with integer coefficients. We show that the Bonahon-Siebenmann decomposition of an achiral and alternating knot is reflected in the Conway polynomial. More explicitly, the Kawauchi conjecture is true for quasi-arborescent knots and counterexamples in the class of alternating knots must be quasi-polyhedral.

Knot polynomials and knot homologies

This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories and give some examples which illustrate a somewhat mysterious correspondence between them.

Adjoint twisted Alexander polynomials of genus one two-bridge knots

We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.

The colored Jones polynomials and the Alexander polynomial of the figure-eight knot

The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by Dehn surgery along the knot. In this paper, we show that for the figure-eight knot the series converges in some cases and the limit equals the inverse of its Alexander polynomial.

Twisted Alexander polynomials of tunnel number one Montesinos knots

We calculate the twisted Alexander polynomials of all tunnel number one Montesinos knots associated to their $SL_2(\mathbb{C})$-representations and obtain their leading coefficients and degrees. As a corollary, we get some interesting examples, that is, nonfibered knots with monic Alexander polynomials which have non-monic twisted Alexander polynomials.

Twisted Alexander Polynomials of $(-2,3,2n+1)$-Pretzel Knots

We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the twisted Alexander polynomials of hyperbolic knots associated to their holonomy representations determine the genus and fiberedness of the knots.

Twisted Alexander polynomials 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 torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.

Cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial

We prove the cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial. We also prove the conjecture for genus one knots with trivial Alexander polynomial, under some additional assumptions.

On lassos and the Jones polynomial of satellite knots

In this master thesis, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot $K$ with Alexander polynomial $Δ_K(t)$, I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial $Δ_K(t^d)$ where $d\in\mathbb{N}\cup \{0\}$. In particular, I prove that if $d\in\{0,1,2,3\}$ these satellite knots have different Jones polynomials. For this purpose, I give rise to a formula for calculating the Jones polynomial of a satellite knot in terms of the Jones polynomials of its pattern and companion.

Various stabilities of the Alexander polynomials of knots and links

In this paper, we study distribution of the zeros of the Alexander polynomials of knots and links in S^3. We call a knot or link "real stable" (resp. "circular stable") if all the zeros of its Alexander polynomial are real (resp. unit complex). We give a general construction of real stable and circular stable knots and links. We also study pairs of real stable knots and links such that the zeros of the Alexander polynomials are interlaced.

The Kauffman Polynomials of 2-bridge Knots

The 2-bridge knots are a family of knots with bridge number 2. In this paper, we compute the Kauffman polynomials of 2-bridge knots using the Kauffman skein theory and linear algebra techniques. Our calculation can be easily carried out using Mathematica, Maple, Mathcad, etc.

Foundations of the Colored Jones Polynomial of singular knots

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.

Polynomial splittings of correction terms and doubly slice knots

We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsváth-Szabó correction terms as smooth double sliceness obstructions vanish for both knots. Recently, Jeffrey Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. As an application, we give a new example of such knots that is distinct from Meier's knots modulo doubly slice knots.

On zeros of the Alexander polynomial of an alternating knot

We prove that for any zero α of the Alexander polynomial of a two-bridge knot, -3 < Re(α) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re(α).

与有限群正则表示相关联的纽结的扭曲Alexander多项式

Twisted Alexander polynomials of knots associated to the regular representations of finite groups

本文研究了与有限群正则表示相关联的纽结扭曲Alexander多项式，给出了通过满射同态与正则表示复合获得的表示下，用Alexander多项式表示的若干公式。

On the Jones polynomial of 2n-plat presentations of knots

In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group $\mathbb{B}_6$ into a group of $5\times 5$ matrices. We also can calculate the Jones polynomial of the $2n$-plat presentations of knots by generalizing the method for the 6-plat presentations of knots.

FK₃纽结多项式

The $\mathrm{FK}_3$ knot polynomial

本文给出与带自同构的12维零散Nichols代数FK₃相关的纽结多项式，计算交叉数≤16的纽结的该多项式，探讨其与李代数相关纽结多项式的差异及其他相关纽结多项式的情况。

Alexander polynomials and signatures of some high-dimensional knots

We give necessary and sufficient conditions for an integer to be the signature of a (4q-1)-knot in the (4q+1)-sphere with a given square-free Alexander polynomial.

多虚拟纽结的多元拧扭多项式

Multivariate writhe polynomial of multi-virtual knots

将拧扭多项式不变量从虚拟纽结推广到多虚拟纽结，并解答了[16]中的若干问题。

Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants

We introduce two sequences of two-variable polynomials $\{ L^n_K (t, \ell)\}_{n=1}^{\infty}$ and $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$, expressed in terms of index value of a crossing and $n$-dwrithe value of a virtual knot $K$, where $t$ and $\ell$ are variables. Basing on the fact that $n$-dwrithe is a flat virtual knot invariant we prove that $L^n_K$ and $F^n_K$ are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using $L^n_K$ we give sufficient conditions when virtual knot does not admit cosmetic crossing change.

Twisted Alexander polynomials of periodic knots

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings. Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley's criterion does n…

A polynomial invariant of virtual knots

The aim of this paper is to introduce a polynomial invariant $f_K(t)$ for virtual knots. We show that $f_K(t)$ can be used to distinguish some virtual knot from its inverse and mirror image. The behavior of $f_K(t)$ under connected sum is also given. Finally we discuss which kind of polynomial can be realized as $f_K(t)$ for some virtual knot $K$.

The $A$-Polynomial and Knot Volume

In this paper, we conjecture a connection between the $A$-polynomial of a knot in $\mathbb{S}^{3}$ and the hyperbolic volume of its exterior $\mathcal{M}_{K}$ : the knots with zero hyperbolic volume are exactly the knots with an $A$-polynomial where every irreducible factor is the sum of two monomials in $L$ and $M$. Herein, we show the forward implication and examine cases that suggest the converse may also be true. Since the $A$-polynomial of hyperbolic knots are known to have at least one irreducible factor which is not the sum of two monomials in $L$ and $M$, this paper considers satellite…

An explicit formula for the $A$-polynomial of the knot with Conway's notation $C(2n, 3)$

An explicit formula for the $A$-polynomial of the knot with Conway's notation $C(2n,3)$ is obtained from the explicit Riley-Mednykh polynomial of it.

具有相同HOMFLYPT多项式的无限纽结族

On infinite knot families with the same HOMFLYPT polynomial

本文研究Kanenobu构造的纽结例子，得到具有相同HOMFLYPT多项式但可被Kauffman多项式区分的新无限纽结族。

Alexander and writhe polynomials for virtual knots

We give a new interpretation of the Alexander polynomial $Δ_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $Δ_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.

Knot Polynomials: Myths and Reality

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.

The Alexander polynomial for Virtual Twist Knots

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $Δ_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from $Δ_0$.

$F$-polynomials of tabulated virtual knots

A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By the construction, $F$-polynomials are generalizations of the Kauffman's Affine Index Polynomial, and are invariants of virtual knot $K$. We present values of $F$-polynomials of oriented virtual knots having at most four classical crossings in a diagram.

The Alexander polynomial of (1,1)-knots

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot, which we call the n-cyclic polynomial. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S^2\times S^1, a result obtained by J. Minkus for 2-bridge knots and extended by the aut…

具有Alexander多项式1和高解结数的光滑切片纽结

Smoothly slice knots with Alexander polynomial 1 and high unknotting number

本文证明存在一个同时具有光滑双切片性、双向手性、Alexander多项式为1且解结数为5的纽结，揭示了这些拓扑性质可以共存。

Slopes and colored Jones polynomials of adequate knots

Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.

Surface pole bracket polynomials of virtual knots and twisted knots

Dye and Kauffman defined surface bracket polynomials for virtual links by use of surface states, and found a relationship between the surface states and the minimal genus of a surface in which a virtual link diagram is realized. They and Miyazawa independently defined a multivariable polynomial invariant of virtual links. This invariant is deeply related to the surface states. In this paper, we introduce the notion of surface pole bracket polynomials for link diagrams in closed surfaces, as a generalization of surface bracket polynomials by Dye and Kauffman. The polynomials induce the invarian…

Twisted Alexander polynomials with the adjoint action for some classes of knots

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in \cite{DHY}.

Some limits of the colored Jones polynomials of the figure-eight knot

We will study the asymptotic behaviors of the colored Jones polynomials of the figure-eight knot. In particular we will show that for certain limits we obtain the volumes of the cone manifolds with singularities along the knot.

Explicit formulae for Chern-Simons invariants of the twist knot orbifolds and Edge polynomials of twist knots

We calculate the Chern-Simons invariants of the twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of the twist knot cone-manifold structures. Following the general instruction of Hilden, Lozano, and Montesinos-Amilibia, we here present the concrete formulae and calculations. We use the Pythagorean Theorem \cite{HMP} to relate the complex length of the longitude and the complex distance between the two axes fixed by two generators. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic twist knot or…

An obstruction to a knot being deform-spun via Alexander polynomials

We show that if a co-dimension two knot is deform-spun from a lower-dimensional co-dimension 2 knot, there are constraints on the Alexander polynomials. In particular this shows, for all n, that not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}.

A kernel of a braid group representation yields a knot with trivial knot polynomials

We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Dehornoy ordering of the braid groups.

A restriction on the Alexander polynomials of $L$-space knots

Using an invariant defined by Rasmussen, we extend an argument given by Hedden and Watson which further restricts which Alexander polynomials can be realized by $L$-space knots.

奇异纽结的扰动 Alexander 多项式

A perturbed Alexander polynomial for singular knots

本文引入奇异纽结与tangle的Alexander多项式版本，通过引入扰动显著增强其功能，并证明对于奇异长结该多项式与先前定义一致。