Journal of Knot Theory and Its Ramifications

In this paper, we determine the second shortest vertical geodesics of cusped hyperbolic manifolds obtained from the Whitehead link. As a corollary, together with the result of [4] this implies that the upper and the lower tunnels for the hyperbolic twist knots are isotopic to the shortest and the second shortest vertical geodesics.

The purpose of this paper is to introduce an infinite sequence {St_k}_k of mutually independent topological invariants of smooth closed plane curves, which is proved to be a natural extension of the rotation number and the strangeness invariant defined by Arnold in [Ar]. We prove a formula to express St_k by using the invariants which are defined in [Oz]. The jumps of St_k at perestroikas of three types (namely cusp point, triple point, and self tangent point perestroika) are investigated, and as a consequence we find that St_k have the order in the sense of Vassiliev equal to 1.

We work in the reduced SU(N, K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet's invariants. Roberts' definition of the Turaev-Viro state sum is exploited. Furthermore, we construct refined Turaev-Viro and Reshetikhin-Turaev TQFTs and study the relationship between them.

The infinite braid group B_∞ admits a left self-distributive structure. In particular, it includes a free monogenerated left self-distributive system, and, therefore, it inherits all properties of the latter object. Here we discuss how such algebraic properties translate into the language of braids. We state new results about braids and propose a list of open questions.

We prove a refinement of Vogel's statement that the Vassiliev invariants of knots coming from semisimple Lie algebras do not generate all Vassiliev invariants. This refinement takes into account the second grading on the Vassiliev invariants induced by cabling of knots. As an application we get an amelioration of the actually known lower bounds for the dimensions of the space of Vassiliev invariants.