Journal of Knot Theory and Its Ramifications

In this paper, we investigate the virtual string links via a probabilistic interpretation. This representation can be used to distinguish some virtual string links from classical string links. In order to study the algebraic structure behind this probabilistic interpretation, we introduce the notion of virtual flat biquandle. The cocycle invariants associated with virtual flat biquandle are also discussed.

In 2011, Bigelow, Ramos and Yi constructed a representation of the group ${{ℬ}}_n$ to subalgebra of planar diagrams $ℂP_n$ of the rook algebra $ℂR_n$. We show that this representation of classical braid groups to the rook algebras can be extended to a representation of the virtual braid groups. This representation is used for constructing invariants of virtual links.

We give characterizations of the skein polynomial for links (as well as Jones and Alexander–Conway polynomials derivable from it), avoiding the usual “smoothing of a crossing” move. As by-products, we have characterizations of these polynomials for knots and for links with any given number of components.

We construct maps from the fundamental groups of complements to some plane configurations to the groups $G_n^k$ for large $k$.

We determine the maximum order of cyclic group actions on the pair $(T^3,{\mathrm{\Sigma }}_g)$ among all embeddings of closed surface ${\mathrm{\Sigma }}_g(g>1)$ into the 3-dimensional torus $T^3$ in the orientable category.

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is called a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynomial from it. For link diagram, one can also define a linear tridle and its presentation matrix. A polynomial invariant can be derived from the matrix just like the Alexander polynomial case.