Journal of Knot Theory and Its Ramifications

Using the Huynh and Lê quantum determinant description of the colored Jones polynomial, we construct a new combinatorial description of the colored Jones polynomial in terms of walks along a braid. We then use this description to show that for a knot which is the closure of a positive braid, the first N coefficients of the Nth colored Jones polynomial are trivial.

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength-critical configurations for knots and links which are different from the ropelength minima for these knot and link types.

We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic surfaces, where some of the former occur as local configurations in degenerated and regenerated surfaces in the latter. In particular, we focus on degenerations which involve intersection points of multiplicity two and three. We demonstrate when the same links arise even when the local arrangements are different.

If A is an abelian quandle and Q is a quandle, the hom set Hom(Q, A) of quandle homomorphisms from Q to A has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom quandle structure. We generalize the result to the case of biquandles, collect observations and results about abelian quandles and the hom quandle, and show that the category of abelian quandles is symmetric monoidal closed.

We analyze various local moves on knot diagrams to show that Jones polynomials must have certain algebraic properties. In particular, we show that the Jones polynomial of a knot cannot be a nontrivial monomial.

We examine some questions regarding which torus links admit colorings by elements of a Dehn quandle.

We describe a "concentration on the diagonal" condition on the Khovanov complex of tangles, show that this condition is satisfied by the Khovanov complex of the single crossing tangles and , and prove that it is preserved by alternating planar algebra compositions. Hence, this condition is satisfied by the Khovanov complex of all alternating tangles. Finally, in the case of 0-tangles, meaning links, our condition is equivalent to a well-known result [E. S. Lee, The support of the Khovanov's invariants for alternating links, preprint (2002), arXiv:math.GT/0201105v1.] which states that the Khovanov homology of a non-split alternating link is supported on two diagonals. Thus our condition is a generalization of Lee's theorem to the case of tangles.