Journal of Knot Theory and Its Ramifications

A word labeled oriented graph (WLOG) is an oriented graph on vertices X = {x₁,…,x_n}, where each oriented edge is labeled by a word in X^±1. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree, are of central importance in view of Whitehead's Asphericity Conjecture. We present a class of aspherical word labeled oriented graphs. This class can be used to produce highly non-injective aspherical labeled oriented trees and also aspherical cyclically presented groups.

In this article we present the following new fact for prime p = 11. For knots 6₂ and 7₂, mincol₁₁ 6₂ = 5 = mincol₁₁ 7₂, along with the following feature. There is a pair of diagrams, one for 6₂ and the other one for 7₂, each of them admitting only non-trivial 11-colorings using five colors, but neither of them admitting being colored with the sets of five colors that color the other one. This new fact is in full contrast with the behavior exhibited by links admitting non-trivial p-colorings over the smaller primes, p = 2, 3, 5 or 7. We also prove results concerning obstructions to the minimization of colors over generic odd moduli. We apply these to finding the right colors to eliminate from non-trivial colorings. We thus prove that 5 is the minimum number of colors for each knot of prime determinant 11 or 13 from Rolfsen's table.

We provide a combinatorial model for spin surfaces. Given a triangulation of an oriented surface, a spin structure is encoded by assigning to each triangle a preferred edge, and to each edge an orientation and a sign, subject to certain admissibility conditions. The behavior of this data under Pachner moves is then used to define a state sum topological field theory on spin surfaces. The algebraic data is a Δ-separable Frobenius algebra whose Nakayama automorphism is an involution. We find that a simple extra condition on the algebra guarantees that the amplitude is zero unless the combinatorial data satisfies the admissibility condition required for the reconstruction of the spin structure.

Utilizing both twisting and writhing, we construct integral tangles with few sticks, leading to an efficient method for constructing polygonal 2-bridge links. Let L be a two-bridge link with crossing number c, stick number s, and n tangles. It is shown that . We also show that if c > 12n + 3, then minimal stick representatives do not admit minimal crossing projections.

Tangles were introduced by J. Conway. In 1970, he proved that every rational 2-tangle defines a rational number and two rational 2-tangles are isotopic if and only if they have the same rational number. So, from Conway's result we have a perfect classification for rational 2-tangles. However, there is no similar theorem to classify rational 3-tangles. In this paper, we introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tangles as well.