Journal of Knot Theory and Its Ramifications

In [3] quandle cohomology is used to produce invariants for particular embeddings of codimension two; 2-cocycles give rise to invariants for (classical) knots and 3-cocycles give rise to invariants for knotted surfaces. This is done by way of a notion of coloring of a diagram. Also, these invariants have the form of the state-sums (or partition functions) used in Statistical Mechanics.

By a careful analysis of these colorings of diagrams we are able to come up with new invariants which correspond to calculation of the partition function at finite temperatures.

The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p + 1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.

We consider a groupoid structure involving graphs and imbedded circles (resp. disks) in a closed orientable surface F (resp. the handleboby H it bounds). We look at generalized connected sums and related operations "induced" by certain morphisms, and introduce a quandle structure involving isotopy classes of imbedded circles and Dehn twists in F. Some relations between these operations are examined, and we look at characterizations of imbedded disks in H arising in this algebraic context.

It has long been known that in high dimensions tbere are examples of irreducible knots which are not prime. Here we show that in fact there are no prime simple knots in high dimensions, with the possible exception of those whose homology is finite. In particular, the result holds for all simple (2q - 1)-knots, q > 1.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

An approach to normal surface theory for non-compact 3-manifolds with respect to ideal pseudo-triangulations is described in [12]. The figure-8 knot complement with a given ideal pseudo-triangulation with two ideal tetrahedra is fully worked out. We construct all Q-fundamental surfaces and the remaining normal surfaces are obtained by linear geometric sums of these. We also construct all almost normal surfaces in the figure-8 knot complement. As a result we obtain an example which does not contain any normal or almost normal surface representing a minimal Seifert surface of the knot. This leads that the figure-8 knot is fibered.