Journal of Knot Theory and Its Ramifications

We construct solutions to the set–theoretic Yang–Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

In this talk, we will give an explicit/combinatorial formulae for Massey–Milnor first non-vanishing linking, and also express this linking in terms of Chern–Simons–Witten perturbative quantum field theory.

A realization of a virtual link diagram is obtained by choosing over/under markings for each virtual crossing. Any realization can also be obtained from some representation of the virtual link. (A representation of a virtual link is a link diagram on an oriented 2-dimensional surface.) We prove that if a minimal genus representation meets certain criteria then there is a minimal genus representation resulting in a knotted realization.

The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro's work on the colored Jones function.

In this paper we prove that two n-gems induce the same manifold if and only if they are linked by a finite sequence of gem moves. A gem move is either a blob move, consisting in the creation or cancellation of an n-dipole, or a clean flip, which is a switch of a pair of edges of the same color that thickens an h-dipole, 1 ≤ h ≤ n - 1, or the inverse operation, which slims an h-dipole, 2 ≤ h ≤ n. Moreover we prove that we can reorder the gem moves, so that all the blob creations precede all clean flips which then precede all the blob cancellations. This reordering is of interest because it is an easy matter to decide whether two gems are linked by a finite sequence of clean flips. As a consequence, if a bound for the number of blob creations is established, then there exists a deterministic finite algorithm to decide whether two gems induce the same manifold or not.

This note gives an explicit calculation of the doubly infinite sequence Δ(p, q, 2m), m ∈ Z of Alexander polynomials of the (p, q) torus knot with m extra full twists on two adjacent strings, where p and q are both positive. The knots can be presented as the closure of the p-string braids , where δ_p = σ_p-1σ_p-2 · σ₂σ₁, or equally of the q-string braids . As an application we give conditions on (p, q) which ensure that all the polynomials Δ(p, q, 2m) with |m| ≥ 2 have at least one coefficient a with |a| > 1. A theorem of Ozsvath and Szabo then ensures that no lens space can arise by Dehn surgery on any of these knots. The calculations depend on finding a formula for the multivariable Alexander polynomial of the 3-component link consisting of the torus knot with twists and the two core curves of the complementary solid tori.

Given a long knot diagram D and a finite quandle Q, we consider the set of all quandle colorings of D with a fixed color q of its initial arc. Using this set we define the family of quandle automorphisms which is a knot invariant. For every element x ∈ Q one can consider the formal sum , taken over all . Such formal sums can be applied to a tangle embedding problem and recognizing non-classical virtual knots.

Let s_Q be the satellite operation on knots defined by a pattern (V, Q), where V is a standard solid torus in S³ and Q ⊂ V is a knot that is geometrically essential in V. It is known (Kuperberg [5]) that if v is any knot invariant of order n ≥ 0, then v ◦ s_Q is also a knot invariant of order ≤ n. We show that if the knot Q has the winding number zero in V, then the satellite map passes n-equivalent knots into (n + 1)-equivalent ones. Kalfagianni [4] has defined for each nonnegative integer n surgery n-trivial knots and studied their properties. It is known that for each n every surgery n-trivial knot is n-trivial. We show that for each n there are n-trivial knots which do not admit a non-unitary n-trivializer that show they to be surgery n-trivial. Przytycki showed [12] that if a knot Q is trivial in S³ and is embedded in V in such a way that it is k-trivial inside V and if a knot is m-trivial, then the satellite knot is (k + m + 1)-trivial. We establish a version of aforementioned Przytycki's result for surgery n-triviality, refining thus a construction for surgery n-trivial knots suggested by Kalfagianni.

The derived group of a permutation representation, introduced by Crowell, unites many notions of knot theory. We survey Crowell's construction, and offer new applications.

The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander polynomials. Also, we extend a well-known theorem of Neuwirth and Stallings giving necessary and sufficient conditions for a knot to be fibered.

Virtual Alexander polynomials provide obstructions for a virtual knot that must vanish if the knot has a diagram with an Alexander numbering. The extended group of a virtual knot is defined, and using it a more sensitive obstruction is obtained.

In this paper, we list in explicit form the factoring relations of the Kauffman bracket skein module (KBSM for short) of a twist knot exterior. This is done using curves decorated by characters of irreducible SL(2, ℂ)-representations. In the process, we exhibit a relation which holds in the KBSM of the knot exterior, called the minimal relation. In the final section we prove that, when specializing the variable of the Kauffman bracket at t = -1, the minimal relation becomes the defining polynomial of the SL(2, ℂ)-character variety of the twist knot.