Journal of Knot Theory and Its Ramifications

Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.

In this paper we investigate the construction of state models for link invariants using representations of the braid group obtained from various gauge choices for a solution of the trigonometric Yang–Baxter equation. Our results show that it is possible to obtain invariants of regular isotopy (as defined by Kauffman) which may not be ambient isotopic. We illustrate our results with explicit computations using solutions of the trigonometric Yang–Baxter equation associated with the one-parameter family of minimal typical representations of the quantum superalgebra U_q[gl(2|1)]. We have implemented MATHEMATICA code to evaluate the invariants for all prime knots up to 10 crossings.

There are some well known indices of knots that give rise to classical knot invariants when minimized for a knot type. By classical knot invariants I mean the minimal number of crossing points, bridge number, braid index, genus and unknotting number. Considering these indices when restricting the minimum over holonomic knots, some of the minima stay the same while others differ. This supplies us with a new set of knot invariants: the holonomic counterparts of the classical knot invariants.

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.

J. S. Carter, S. Kamada and M. Saito showed that there is one to one correspondence between the virtual Reidemeister equivalence classes of virtual link diagrams and the stable equivalence classes of link diagrams on compact oriented surfaces. Using the result, we show how to obtain the supporting genus of a projected virtual link by a geometric method. From this result, we show that a certain virtual knot which cannot be judged to be non-trivial by known algebraic invariants is non-trivial, and we suggest to classify the equivalence classes of projected virtual links by using the supporting genus.

There are obvious inequalities relating the Nakanishi index of a knot, the bridge number, the degree 2n of the Alexander polynomial and the length of the chain of Alexander ideals. We give examples for every positive value of n to show that these bounds are sharp.

Let K be a two-bridge knot in S³. Then K is also denoted as the four-plat, b(p,q) to indicate its association with some rational number p/q.

The lens space L = L(p,q) admits an isometry, τ, of order 2, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K.

In this paper, the isometries of these orbifolds are classified; this is equivalent to computing the isometries of (S³,K).

A general result of Epstein and Thurston implies that all link groups are automatic, but the proof provides no explicit automaton. Here we show that the groups of all torus links are groups of fractions of so-called Garside monoids, i.e., roughly speaking, monoids with a good theory of divisibility, which allows us to reprove that those groups are automatic, but, in addition, gives a completely explicit description of the involved automata, thus partially answering a question of D. F. Holt.

We define a type of Stallings twists, which represents a "complexity" of the twist, and show that the Stallings twists of a certain type on a fiber surface for a fibered link in S³ can be realized by plumbing one Hopf band and deplumbing another Hopf band. Using this technique, we construct stable Hopf plumbings which are not Hopf plumbings with an arbitrary high first Betti number.