Journal of Knot Theory and Its Ramifications

Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous cycles. The methods are applied to prove that boundary homomorphisms in a homology exact sequence vanish.

We study the variation of the Tait number of a closed space curve according to its different projections. The results are used to compute the writhe of a knot, leading to a closed formula in case of polygonal curves.

We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces to that of the base space. Finally, we show that this places restrictions on the number of finite and cyclic Dehn fillings of the knot complements in the covering spaces beyond those imposed by Culler-Gordon-Luecke-Shalen and Boyer-Zhang.

This paper concerns 3-manifolds X obtained by Dehn surgery on a knot in S³, in particular those which contain embedded projective planes. Either, they are homeomorphic to the 3-real projeclive space ℝP³, or they are reducible.

Let p be the number of intersections of a projective plane in X with the core of the solid torus added during surgery. We prove here that either X is reducible or p is bigger than or equal to five.

Consequently, if X is homeomorphic to ℝP³ then all its projective planes are pierced at least in five points by the core of the surgery. This result is considered as a step towards showing that ℝP³ cannot be obtained by a Dehn surgery along a knot in S³.

In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.

We deal with the properties of derivative of Jones polynomial for algebraically split links. In particularly, the properties of ν_n(L) and ϕ_n(L) are discussed.

A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is computed.