Journal of Knot Theory and Its Ramifications

We give the counterexamples for a conjecture of Fiedler concerning the ribbon genus of a link.

In this note, we will give an approach to determine the unknotting number by a surgical view of Alexander matrix.

Let f: K² → M³ be an embedding of a compact, connected 2-complex into a compact, connected, orientable 3-manifold. We are interested to know, to which extent K² determines (up to homeomorphism) its regular neighborhood N(f(K²)) ⊆ M³. We exhibit a list of four obstructions to uniqueness, and prove that in the absence of these, the regular neighborhoods are uniquely determined: Let f,K²,M³ be as above and assume M³ is prime and not a Poincaré-counterexample. If N(f(K²)) does not contain essential annuli and has connected boundary, then N is determined by K².

We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain index conditions the link invariants colored by the representations of coset factorize into the products of the the link invariants colored by the representations of the two groups in the coset. But the 3-manifold invariants do not behave so simply in general due to the nontrivial branching rules of the coset. Examples in the parafermion cosets and diagonal cosets show that 3-manifold invariants of the coset may be finer than the products of the 3-manifold invariants associated with the two groups in the coset, and these two invariants do not seem to be simply related in some cases, for an example, in the cases when there are issues of "fixed point resolutions". In the later case our framework provides a representation theory understanding of the underlying unitary modular categories which has not been obtained before.

In this paper we prove that if M_K is the complement of a non-fibered twist knot K in , then M_K is not commensurable to a fibered knot complement in a ℤ/2ℤ-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by Calegari and Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

In earlier works (by Arnold and the author) the approach of singularity theory was used to construct invariants of degree 1 of plane immersed curves. This provides a finer classification of these immersions. In this paper, we construct higher degree invariants by "integrating" some of the basic invariants found earlier of almost generic immersed curves.

The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n + 1], the quantum dimension of the (n + 1)-dimensional irreducible representation of quantum , and the other in which it evaluates to 1. For each normalization we construct a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic.