Journal of Knot Theory and Its Ramifications

We introduce a ribbon presentation to describe a ribbon knot, and equivalences between them. Superspun knots of classical knots are well known to be ribbon knots, and we give a construction of ribbon presentations of the knots by means of classical knot diagrams. We can then extend the classical Reidemeister moves into higher dimensions, and we prove that the presentations of a superspun knot are stably equivalent by using these moves.

Four approaches to construct polynomial invariants for trivalent knotted graphs in S³ are compared. The first approach is based on vertex model with R-matrices and Glebsh-Gordan coefficients, appearing in SL_q(2)-representations theory, as Boltzman weights. The second approach is based on Kauffman's quantum spin network theory, the third one is based on Witten-Turaev area-coloring model (or face model) based on quantum 6j-symbols, where q is root of unity. The fourth approach is based on the same face (or area-coloring) model, but q is not root of unity. The coincidence (up to certain normalization) of topological invariants, arising from these four state models, is proved.

If X is the space of the trefoil knot, we can find, for every integer e, two slopes on ∂X, ρ_e, γ_e, such that the Dehn fillings X (ρ_e) and X (γ_e) are homeomorphic 3-manifolds. The cores of the surgeries are inequivalent knots with homeomorphic complements. Analogous results are available if X is the knotspace of a 2-bridge torus knot. In an atoroidal homology sphere knots are determined by their complement if and only if hyperbolic knots are.

We prove that a certain geometric construction on links in the 3-sphere preserves the properties: non-trivial, non-split, non-boundary.

The problem of finding the genus of a given link type of braid index 3 is solved by constructive methods. The results depend upon a new solution to the conjugacy problem in B₃. In this solution, conjugacy classes are represented by shortest words in terms of cyclically symmetric elementary braids which are used as generators in the new presentation of B₃. By related results of Bennequin [2] and of Birman and Menasco [4], the minimal spanning surfaces are described by these shortest words. An effective algorithm is given to find these shortest words starting with an arbitrary 3-braid representative of the link. It is proved that up to an easily described family of exceptional cases, a link of braid index 3 has a unique isotopy class of minimal spanning surfaces. The growth functions of B₃ and its conjugacy classes are also given here.