Related papers: Extension of KNTZ trick to non-rectangular represe…
We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order…
We propose an infinitesimal counterpart to the notion of braided category. The corresponding infinitesimal braidings are natural transformations which are compatible with an underlying braided monoidal structure in the sense that they…
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…
N. Katz has shown that any irreducible representation of the Galois group of F_q((t)) has unique extension to a special representation of the Galois group of k(t) unramified outside 0 and infinity and tamely ramified at infinity. In this…
We consider braids with repeating patterns inside arbitrary knots which provides a multi-parametric family of knots, depending on the "evolution" parameter, which controls the number of repetitions. The dependence of knot (super)polynomials…
The recent proof by Bigelow and Krammer that the braid groups are linear opens the possibility of applications to the study of knots and links. It was proved by the first author and Menasco that any closed braid representative of the unknot…
We study a class of representations of the Cuntz algebras O_N, N=2,3,..., acting on L^2(T) where T=R/2\pi Z. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into…
A branched twist spin is a generalization of twist spun knots, which appeared in the study of locally smooth circle actions on the $4$-sphere due to Montgomery, Yang, Fintushel and Pao. In this paper, we give a sufficient condition to…
We show a correspondence between simple continued fraction expansions of irrational numbers and irreducible permutative representations of the Cuntz algebra ${\cal O}_{\infty}$. With respect to the correspondence, it is shown that the…
We study the mixed-integer epigraph of a special class of convex functions with non-convex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are…
Empirical analysis of many colored knot polynomials, made possible by recent computational advances in Chern-Simons theory, reveals their stability: for any given negative N and any given knot the set of coefficients of the polynomial in…
Various properties of a class of braid matrices, presented before, are studied considering $N^2 \times N^2 (N=3,4,...)$ vector representations for two subclasses. For $q=1$ the matrices are nontrivial. Triangularity $(\hat R^2 =I)$…
We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or…
The study of a certain class of matrix integrals can be motivated by their interpretation as counting objects of knot theory such as alternating prime links, tangles or knots. The simplest such model is studied in detail and allows to…
By proving a connected sum formula for the Legendrian invariant $\lambda_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots.
In this paper, we build on the biquasiles and dual graph diagrams introduced in arXiv:1610.06969. We introduce \textit{biquasile Boltzmann weights} that enhance the previous knot coloring invariant defined in terms of finite biquasiles and…
The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length…
We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally…
We consider a twisted version of quantum groups corepresentations. This generalization amounts to include in the theory the case where quantum space coordinates and its endomorphism matrix entries belong to a non-commutative quadratic…
This work is dedicated to the consideration of the construction of a representation of braid group generators from vertex models with $N$-states, which provides a great way to study the knot invariant. An algebraic formula is proposed for…