This paper introduces a self-linking invariant for virtual knots and links, and relates this invariant to a state model called the binary bracket, and to a class of coloring problems for knots and links that include classical coloring problems for cubic graphs.
This is a short summary of a topological model of site-specific recombination, a cellular reaction that creates knots and links out of circular double stranded DNA molecules. The model is used to predict and characterise the topology of the products of a reaction on double stranded DNA twist knots. It is shown that all such products fall into a small family of Montesinos knots and links, meaning that the knot and link type of possible products is significantly reduced, thus aiding their experimental...
We consider a cobordism category whose morphisms are punctured connected sums of ’s (wormhole spaces) with embedded admissibly colored banded trivalent graphs. We define a TQFT on this cobordism category over the field of rational functions in an indeterminant A. For r large, we recover, by specializing A to a primitive 4rth root of unity, the Witten-Reshetikhin-Turaev TQFT restricted to links in wormhole spaces. Thus, for r large, the rth Witten-Reshetikhin-Turaev invariant of a link in some wormhole...
We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group.
Motivated by the work of Chmutov, Duzhin and Lando on Vassiliev invariants, we define a polynomial on weighted graphs which contains as specialisations the weighted chromatic invariants but also contains many other classical invariants including the Tutte and matching polynomials. It also gives the symmetric function generalisation of the chromatic polynomial introduced by Stanley. We study its complexity and prove hardness results for very restricted classes of graphs.
In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard Artin presentation for generalizations of braids. Namely, we consider presentations with small number of generators, Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V.~V.~Chaynikov on the word and conjugacy problems for the singular...
The affine Birman-Wenzl-Murakami algebras can be defined algebraically, via generators and relations, or geometrically as algebras of tangles in the solid torus, modulo Kauffman skein relations. We prove that the two versions are isomorphic, and we show that these algebras are free over any ground ring, with a basis similar to a well known basis of the affine Hecke algebra.
The Alexander ideals of classical knots are characterised, a result which extends to certain higher dimensional knots.