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We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on…
In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write…
It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we…
Let $p$ be a prime. In this short note we study some families of super congruences involving the following alternating sums \begin{equation*} \sum_{\substack{j_1+j_2+\cdots+j_n=2 p^r p\nmid j_1 j_2 \cdots j_n}}…
In this paper, we define a braid invariant, the purified determinant $P(b)$ of a braid $b$, considering the determinant of the crossing matrix of a pure braid derived from $b$, and show that $P(b_1 b_2)=P(b_2 b_1)$ for any pair of…
We derive new formulas for the Jones polynomial and the Kauffman bracket polynomial of a rational link represented by a standard diagram that is not necessarily alternating. These formulas generalize the results of Qazaqzeh, Yasein, and…
We define an integer valued invariant for two-component links in S^3 by counting projective SU(2) representations of the link group having non-trivial second Stiefel-Whitney class. We show that our invariant is, up to sign, the linking…
Complete non-ambiguous trees have been studied in various contexts. Recently, a conjecture was made about their determinants, and subsequently proved by Aval. An alternative proof is given here.
We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form $T((p,q),(2,s))$ where $p$ and $q$ are coprime and $s$ is nonzero. When $s = 2n$, these links are the twisted torus knots…
The Jones unknot conjecture states that the Jones polynomial distinguishes the unknot from nontrivial knots. We prove it for knots up to 23 crossings.
An alternating colouring function is defined on strings over the alphabet $\{0, 1\}$. It divides the strings in colourable and non-colourable ones. The points in the subshift of finite type defined by forbidding all non-colourable strings…
We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…
We use Feynman diagrams to prove a formula for the Jones polynomial of a link derived recently by N.~Reshetikhin. This formula presents the colored Jones polynomial as an integral over the coadjoint orbits corresponding to the…
In this paper we review the definitions of homogeneous and alternative links. We also give two new characterizations of an alternative link diagram, one within the context of the enhanced checkerboard graph and another from the labeled…
This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…
To each link $L$ in $S^3$ we associate a collection of certain labelled directed trees, called width trees. We interpret some classical and new topological link invariants in terms of these width trees and show how the geometric structure…
The alternating-runs polynomial enumerates alternating runs in the symmetric group. There are three formulae for the number of permutations, $R_{n,k}$ in $\mathfrak{S}_n$ with $k$ alternating runs, but all of them are complicated. We show…
Using the colored Kauffman skein relation, we study the highest and lowest $4n$ coefficients of the $n^{th}$ unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the…
We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a…
In this paper, we introduce two families of planar and self-similar graphs which have small-world properties. The constructed models are based on an iterative process where each step of a certain formulation of modules results in a final…