Alexander Polynomial Research Articles

A Note on the Concordance Z-Genus

We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden, Livingston, and Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.

Derivatives with Alexander pairs for quandles

Twisted Alexander invariants for finitely presented groups equipped with a linear representation are defined using free derivatives introduced by R. H. Fox. In this paper, we define derivatives for quandles and use them to introduce new invariants for fin

Branched covers bounding rational homology balls

Prime power fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. In this paper, we give a new construction of non-slice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander polynomials, and we introduce new techniques to simplify their calculation.

Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

We have recently proposed [1] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with SU(N) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots T[2,2k+1]. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank N and the representation R.

A Characterization of T2g+1,2 among Alternating Knots

Let K be a genus g alternating knot with Alexander polynomial \({\Delta _K}(T) = \sum\nolimits_{i = - g}^g {{a_i}{T^i}}\). We show that if |ag| = |ag−1|, then K is the torus knot T2g+1,±2. This is a special case of the Fox Trapezoidal Conjecture. The proof uses Ozsváth and Szabó’s work on alternating knots.

A basis for the Kauffman skein module of the product of a surface and a circle

The Kauffman bracket skein module $S(M)$ of a 3-manifold $M$ is a $\mathbb{Q}(A)$-vector space spanned by links in $M$ modulo the so-called Kauffman relations. In this article, for any closed oriented surface $\Sigma$ we provide an explicit spanning family for the skein modules $S(\Sigma\times S^1)$. Combined with earlier work of Gilmer and Masbaum, we answer their question about the dimension of $S(\Sigma\times S^1)$ being $2^{2g+1} + 2g -1$.

Local-move identities for the ℤ[t,t−1]-Alexander polynomials of 2-links, the alinking number, and high-dimensional analogues

A well-known identity [Formula: see text] holds for three 1-links [Formula: see text], [Formula: see text], and [Formula: see text] which satisfy a famous local-move relation, where [Formula: see text] becomes the Alexander–Conway polynomial of [Formula: see text] if we let [Formula: see text]. We prove a new local-move identity for the [Formula: see text]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander–Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the [Formula: see text]-Alexander polynomials of 2-links and the alinking number of 2-links. We produce high-dimensional analogues of these results. Furthermore, we prove that in the 2-dimensional case we cannot normalize the [Formula: see text]-Alexander polynomials to be compatible with our identity but that in a high-dimensional case we can do that to be compatible with our new identity.

Forty years in mathematics: Personal perspective (Part I)

Conference like this in Dallas is a perfect occasion to give some reminiscences of my 40 years of work as a mathematician, and to muse a little about my mathematical past. It is a rare occasion to be a little sentimental. In this essay, I summarize and expand my birthday talk given at University of Texas at Dallas conference. There, I mostly covered the early years of my mathematical career ending with a detailed description about the smallest volume hyperbolic 3-manifold and the HOMFLYPT polynomial. Thus I seldom venture to the time after 1984 (for example to mention skein modules constructed in April 1987) and only occasionally after 1995 when I settled to work at George Washington University and have had several successful PhD students.

Homflypt skein theory, string topology and 2-categories

We show that relations in Homflypt type skein theory of an oriented [Formula: see text]-manifold [Formula: see text] are induced from a [Formula: see text]-groupoid defined from the fundamental [Formula: see text]-groupoid of a space of singular links in [Formula: see text]. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental [Formula: see text]-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.

Surgeries, sharp 4–manifolds and the Alexander polynomial

Work of Ni and Zhang has shown that for the torus knot $T_{r,s}$ with $r>s>1$ every surgery slope $p/q \geq \frac{30}{67}(r^2-1)(s^2-1)$ is a characterizing slope. In this paper, we show that this can be lowered to a bound which is linear in $rs$, namely, $p/q\geq \frac{43}{4}(rs-r-s)$. The main technical ingredient in this improvement is to show that if $Y$ is an $L$-space bounding a sharp 4-manifold which is obtained by $p/q$-surgery on a knot $K$ in $S^3$ and $p/q$ exceeds $4g(K)+4$, then the Alexander polynomial of $K$ is uniquely determined by $Y$ and $p/q$. We also show that if $p/q$-surgery on $K$ bounds a sharp 4-manifold, then $S^3_{p'/q'}(K)$ bounds a sharp 4-manifold for all $p'/q'\geq p/q$.

Embedding spheres in knot traces

AbstractThe trace of the$n$-framed surgery on a knot in$S^{3}$is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded$2$-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable$3$-dimensional knot invariants. For each$n$, this provides conditions that imply a knot is topologically$n$-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

Sliding dynamics of ring chain on a knotted polymer in rotaxane

Rotaxane is a supramolecule in which ring chains are threaded on an axial chain, showing a fascinating sliding dynamics of ring chains along the axial chain. In this study, we investigated the sliding dynamics of ring chains on a knotted polymer chain in rotaxane, exhibiting a sub-diffusion at the intermediate time range resulting from the intra-strand interaction between the ring chains and the knotted axial chain in the knot crossing region. Meanwhile, the non-monotonous dependence of sliding diffusion coefficient D of ring chains on ring size Nring indicates that there exist two competing effects: one is the friction effect between the ring chains and the axial chain in which the ring chains slide fast along the knotted axial chain for large Nring, and another is the knot effect in which the ring chains move slowly along the knotted axial chain for large Nring because the knot crossing hinders ring chains to pass through for large Nring. Based on the unique sliding sub-diffusion behaviour of the threaded ring chains, we also proposed a new approach to estimate the average knot size of the knotted axial chain, and our estimation about the average knot size is in good agreement with the result according to the Alexander polynomial calculation using the free python package TOPOLY. These results may provide a fundamental view for understanding the dominant factors of the sliding dynamics of ring chains in complex polymer-based rotaxane.

Knot probabilities in equilateral random polygons

We consider the probability of knotting in equilateral random polygons in Euclidean three-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal scaling formula for the knotting probability with the number of edges. This scaling formula involves an exponential function, independent of knot type, with a power law factor that depends on the number of prime components of the knot. The unknot, appearing as a composite knot with zero components, scales with a small negative power law, contrasting with previous studies that indicated a purely exponential scaling. The methodology incorporates several improvements over previous investigations: our random polygon data set is generated using a fast, unbiased algorithm, and knotting is detected using an optimised set of knot invariants based on the Alexander polynomial.

Faithfullness of Geometric Action of Skein Algebras

We show that the action of the Kauffman bracket skein algebra of a surface Σ on the skein module of the handlebody bounded by Σ is faithful if and only if the quantum parameter is not a root of 1.

Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds

We study the Smith forms of matrices of the form f(Cg) where f(t),g(t)∈R[t], where R is an elementary divisor domain and Cg is the companion matrix of the (monic) polynomial g(t). Prominent examples of such matrices are circulant matrices, skew-circulant matrices, and triangular Toeplitz matrices. In particular, we reduce the calculation of the Smith form of the matrix f(Cg) to that of the matrix F(CG), where F,G are quotients of f(t),g(t) by some common divisor. This allows us to express the last non-zero determinantal divisor of f(Cg) as a resultant. A key tool is the observation that a matrix ring generated by Cg – the companion ring of g(t) – is isomorphic to the polynomial ring Qg=R[t]/<g(t)>. We relate several features of the Smith form of f(Cg) to the properties of the polynomial g(t) and the equivalence classes [f(t)]∈Qg. As an application we let f(t) be the Alexander polynomial of a torus knot and g(t)=tn−1, and calculate the Smith form of the circulant matrix f(Cg). By appealing to results concerning cyclic branched covers of knots and cyclically presented groups, this provides the homology of all Brieskorn manifolds M(r,s,n) where r,s are coprime.

Moves relating C-complexes: A correction to Cimasoni's “A geometric construction of the Conway potential function”

In work from 2004, Cimasoni gave a geometric computation of the multivariable Conway potential function in terms of a generalization of a Seifert surface for a link called a C-complex [2]. These surfaces were introduced by Cooper [9,10] for 2-component links and were used to compute Alexander invariants, signatures and nullities. Cooper presents a sequence of geometric moves which relate any two C-complexes for a 2-component link. Cimasoni gives an extension of Cooper's moves, providing a family of moves which relates any two C-complexes for a fixed link. This allows for an approach to studying links from the point of view of C-complexes and in following papers it has been used to derive invariants. Cimasoni's result is incorrect. We present counterexamples, a correction with detailed proof, and check that subsequent results relying on this lemma still hold.

On the genus two skein algebra

We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the $t=q$ specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.

Alexander polynomial and spanning trees

Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander polynomial evaluated at [Formula: see text] gives the weighted number of the spanning trees of the graph.

Virtual parity Alexander polynomial

In this paper, we define the virtual Alexander polynomial following the works of Boden et al. (2016) [Alexander invariants for virtual knots, J. Knot Theory Ramications 24(3) (2015) 1550009] and Kaestner and Kauffman [Parity biquandles, in Knots in Poland. III. Part 1, Banach Center Publications, Vol. 100 (Polish Academy of Science Mathematical Institute, Warsaw, 2014), pp. 131–151]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots cannot be unknotted by crossing changes on only odd crossings.

Low-Dimensional Topology and Number Theory

Because of the pandemia, the workshop on “Low-Dimensional Topology and Number Theory” could not be realized in the usual format or in the new hybrid format. Instead, a subgroup consisting of 6 participants used the week at the MFO mostly for informal discussions, collaborations and research in the topic of the workshop, including some online contacts to other participants who were not able to come to the MFO. At the institute talks were given by Campbell Wheeler on “WRT invariants &amp; quantum modularity”, Gregor Masbaum on “Generic skein modules”, Gaëtan Borot on “Counting multicurves on surfaces with respect to hyperbolic vs. combinatorial geometry”, Roland van der Veen on “1-cocycle invariants of knots”, and by Michael Ontiveros on “From a triangulated lens space to a modular form”.