Journal of Knot Theory and Its Ramifications

We show that the twisted signature invariants of boundary link concordance derived from unitary representations of the free group are actually invariants of ordinary link concordance. We also show how the discontinuity locus of this signature function is determined by Seifert matrices of the link.

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

We introduce a geometric construction, called a clasp move, which is used to prove several results on stick numbers of links. In particular, we show that all two-component six-crossing links have stick number 8, produce small crossing number knots that can be constructed with fewer sticks than crossings, and improve known upper bounds for the stick number of certain classes of links.

We define a knot invariant and a 2-knot invariant from any finite categorical group. We calculate an explicit example for the Spun Trefoil.

We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.

Motivated by Massey–Milnor linking and Chern–Simon–Witten perturbative quantum field theory, we developed some combinatorial and diagrammatic study in this paper, aiming at knot theory in the combinatoric aspect.