Related papers: Infinitely many two-variable generalisations of th…

Building further on work of Marin and Wagner, we give a cubic braid-type skein theory of the Links--Gould polynomial invariant of oriented links and prove that it can be used to evaluate any oriented link, adding this polynomial to the list…

For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this…

We consider the factorization of permutations into bandwidth 1 permutations, which are products of mutually nonadjacent simple transpositions. We exhibit an upper bound on the minimal number of such factors and thus prove a conjecture of…

We introduce a new version $kk^{\rm alg}$ of bivariant $K$-theory that is defined on the category of all locally convex algebras. A motivating example is the Weyl algebra $W$, i.e. the algebra generated by two elements satisfying the…

We use a trivial delta method with multiplicative characters for congruence detection to prove the Weyl bound for GL(2) in $t$-aspect for a holomorphic or Hecke-Maass cusp form of arbitrary level and nebentypus. This parallels the work of…

We present a short and unified representation-theoretical treatment of type A link invariants (that is, the HOMFLY-PT polynomials, the Jones polynomial, the Alexander polynomial and, more generally, the gl(m|n) quantum invariants) as link…

We study generalizations of a classical link invariant -- the multivariable Alexander polynomial -- to tangles. The starting point is Archibald's tMVA invariant for virtual tangles which lives in the setting of circuit algebras, and whose…

We extend the polynomial Pell's equation satisfied by univariate Chebyshev polynomials on [--1, 1] from one variable to several variables, using orthogonal polynomials on regular domains that include cubes, balls, and simplexes of arbitrary…

We consider a translation and dilation invariant system consisting of k diagonal equations of degrees 1,2,...,k with integer coefficients in s variables, where s is sufficiently large in terms of k. We show via the Hardy-Littlewood circle…

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 $\delta$--graded knot…

This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is…

We study an invariant, the secondary trace, attached to two commuting endomorphisms of a 2-dualizable object in a symmetric monoidal higher category. We establish a secondary trace formula which encodes the natural symmetries of this…

Let $\Gamma$ be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of $\Gamma$ into $\mathrm{SL}_n(\mathbf{C})$ which are the sum of two irreducible representations. For such…

This paper contains the strongest and at the same time most calculable knot invariant ever. Let $\Theta$ be the topological moduli space of all ordered oriented tangles in 3-space. We construct a non-trivial combinatorial 1-cocycle…

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical…

We construct new invariant polynomial for long virtual knots. It is a generalization of Alexander polynomial. We designate it by $\zeta$ meaning an analogy with $\zeta$-polynomial for virtual links. A degree of $\zeta$-polynomial estimates…

The maximum length of the shortest path from a leaf to the root of a skein tree for knots and links gives a measure of the complexity of computing link polynomials by the skein relation (the Jones polynomial, the Alexander-Conway…

This paper describes a method to obtain state model parameters for an infinite series of Links-Gould link invariants LG^{m,n}, based on quantum R matrices associated with the (\dot{0}_m | \dot{\alpha}_n) representations of the quantum…

We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as fixed…

We generalize the basic results of Vinberg's \theta-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard)…