Narrow Results

We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms of linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a simple q-series for the tail of the knot 8₅. Furthermore, we use this expansion to obtain an easy determination of the theta coefficients.

We derive a minimal generating set of planar moves for diagrams of surfaces embedded in the four-space. These diagrams appear as the bonded classical unlink diagrams.

We create an invariant of virtual $Y$-oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson–Pico, and Graves-Nelson-T. We computed all tribrackets, Niebrzydowski algebras and virtual Niebrzydowski algebras of orders 3 and 4, and provide generative code for all data sets.

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut(X) acts transitively on the vertices and the vertex-stabilizer A_v of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.