Journal of Knot Theory and Its Ramifications

The presentation of link polynomials, arising from representations of quantum group SL_q(2) by SL_q(2)-spin networks is given. The explicit form of cabling formula for these polynomials is written. The connection between 6j-symbols in q-spin network theory and Rakah-Wigner q-6j-symbols is shown. The Kauffman's hypothesis about coincidence of his 3-manifold invariants and Turaev-Viro invariants for 3-manifolds is proved.

Any closed oriented surface embedded in R⁴ is described as a closed 2-dimensional braid and its braid index is defined. We study 2-dimensional braids through a method to describe them using graphs on a 2-disk and show that braid index two surfaces in R⁴ are unknotted and braid index three surfaces are ribbon.

We define a new hierarchy of isotopy invariants of colored oriented links through oriented tangle diagrams. We prove the colored braid relation and the Markov trace property explicitly.

Canonical triangulations provide an algorithm to compute the symmetry groups of hyperbolic links and their complements, along with information on how each symmetry acts on the link components. The latter information determines the chirality and invertibility of the link. Symmetry groups of knots to ten crossings and multicomponent hyperbolic links to nine crossing are listed, with some exceptions. A Macintosh computer program, available free of charge, computes symmetry groups interactively.

We answer a question of Jones concerning the positivity of the two-variable (Homfly) knot polynomial P_L when L is a positive link. We show that, in this case, P_L (v, z) is a positive polynomial in z when v ∈ (0, 1).

We solve the planar generalizations of the Yang-Baxter equation using Kauffman's bracket methods. These solutions are applied to get representations of certain Coxeter groups that are quotients of generalized braid groups.