The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the present paper is a generalization of Conway's mutation of knots and links. Instead of flipping a 2-strand tangle, one flips a many-string tangle to produce a generalized mutant. In the presence of rotational symmetry in that tangle, the result is called a "rotant". We show that if a rotant is sufficiently simple, then its Jones polynomial agrees with that of the original link. As an application, this provides a method of generating many examples of links with the same Jones polynomial, but different Alexander polynomials. Various other knot polynomials, as well as signature, are also invariant under such moves, if one imposes more stringent conditions upon the symmetries. Applications are also given to polynomials of satellites and symmetric knots.
For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.
Given a hyperbolic knot, we prove that the Reidemeister torsion of any lift of the holonomy to SL(2,C) is invariant under mutation along a Conway sphere.
Kearton observed that mutation can change the concordance class of a knot. A close examination of his example reveals that it is of 4-genus 1 and has a mutant of 4-genus 0. The first goal of this paper is to construct examples to show that for any pair of nonnegative integers m and n there is a knot of 4-genus m with a mutant of 4-genus n. A second result of this paper is a crossing change formula for the algebraic concordance class of a knot, which is then applied to prove the invariance of the algebraic concordance class under mutation. The paper concludes with an application of crossing cha…
Let $\mathcal{L}$ be a knot with a fixed positive crossing and $\mathcal{L}_n$ the link obtained by replacing this crossing with $n$ positive twists. We prove that the knot Floer homology $\widehat{\text{HFK}}(\mathcal{L}_n)$ `stabilizes' as $n$ goes to infinity. This categorifies a similar stabilization phenomenon of the Alexander polynomial. As an application, we construct an infinite family of prime, positive mutant knots with isomorphic bigraded knot Floer homology groups. Moreover, given any pair of positive mutants, we describe how to derive a corresponding infinite family positive mutan…
Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional linear representations of the fundamental group of X. This provides a simpler and faster algorithm to compute these twisted polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3,7,9,11,15), corre…
In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp shapes. The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which …
In an earlier paper, we introduced a collection of graded Abelian groups $\HFKa(Y,K)$ associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the Kinoshita-Terasaka knots and their ``Conway mutants''. These results show that $\HFKa$ contains more information than the Alexander polynomial and the signature of these knots; and they also illustrate the fact that $\HFKa$ detects mutation. We also calculate $\HFKa$ for certain pretzel knots, and knots with small crossing number ($n\leq 9$). Our calculatio…
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 be achieved by a sequence of at most two genus 2 mutations, the invariants that are preserved by genus 2 mutation are a proper subset of those preserved by Conway mutation. In particular, while the Alexander and Jones polynomials are preserved by genus 2 mutation, the HOMFLY-PT p…
We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two. The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for n>1 there is an isomorphism phi from the concordance group C_n of knotted (2n-1)-spheres in S^{2n+1} to an algebraically defined group G_{+-}; furthermore, G__{+-} is isomorphic to the infinite direct sum Z^infty direct sum Z_2^infty direct sum Z_4^infty. It was a startling consequence of the work of Casson and Gordon that in the classical ca…
We give a new, elementary proof that Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$--coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that $δ$--graded knot Floer homology is mutation--invariant. Using the Clifford module structure on $\widetilde{\text{HFK}}$ induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let $L'$ be a link obtained from $L$ by mutating the tangle $T$. Suppose some rational closure of $T$ corresponding to the mutation is the unlink on any number of compon…
It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtration and a spectral sequence that converges to the reduced Khovanov homology of K. We show that the E_2-term of this spectral sequence is a matroid invariant and hence invariant under mutation.
This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a Type 2 Vassiliev Invariant coming from the polynomial.
It is known that the first two-variable Links--Gould quantum link invariant $LG\equiv LG^{2,1}$ is more powerful than the HOMFLYPT and Kauffman polynomials, in that it distinguishes all prime knots (including reflections) of up to 10 crossings. Here we report investigations which greatly expand the set of evaluations of $LG$ for prime knots. Through them, we show that the invariant is complete, modulo mutation, for all prime knots (including reflections) of up to 11 crossings, but fails to distinguish some nonmutant pairs of 12-crossing prime knots. As a byproduct, we classify the mutants with…
We show that the zeroth coefficient of the cables of the HOMFLY polynomial (colored HOMFLY polynomials) does not distinguish mutants. This makes a sharp contrast with the total HOMFLY polynomial whose 3-cables can distinguish mutants.
The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, and Kaufman two-variable polynomial, Khovanov homology, factorizability of the polynomials, and knot primeness detection.
We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type.
We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. In this paper we show that every torus knot $T(p,q)$ is detected by its knot Floer homology and $A$-polynomial. We also give a one-parameter family of infinitely many hyperbolic knots in $S^3$ each of which is detected by its knot Floer homology and $A$-polynomial. In addition we give a cabling formula for the A-polynomials of cabled knots in $S^3$, which is of independent interest. I…
For knots in $S^3$, it is well-known that the Alexander polynomial of a ribbon knot factorizes as $f(t)f(t^{-1})$ for some polynomial $f(t)$. By contrast, the Alexander polynomial of a ribbon $2$-knot is not even symmetric in general. Via an alternative notion of ribbon $2$-knots, we give a topological condition on a $2$-knot that implies the factorization of the Alexander polynomial.
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.
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.
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.
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.
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.
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…
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.
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.
We present a notion of mutation of hyperbolic polyhedra, analogous to mutation in knot theory, and then present a general question about commensurability of mutant pairs of polyhedra. We motivate that question with several concrete examples of mutant pairs for which commensurability is unknown. The polyhedra we consider are compact, so techniques involving cusps that are typically used to distinguishing mutant pairs of knots are not applicable. Indeed, new techniques may need to be developed to study commensurability of mutant pairs of polyhedra.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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).
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.
International Center for Mathematics, Department of Mathematics Southern University of Science and Technology（南方科技大学数学中心、数学系）；Department of Mathematics University of Illinois（伊利诺伊大学数学系）
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.
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.
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…
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 $K$.
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…
