Narrow Results

We introduce Niebrzydowski algebras, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for $Y$-oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of $Y$-oriented Reidemeister moves. We give some examples to demonstrate that the counting invariant can distinguish some $Y$-oriented trivalent spatial graphs and handlebody-links.

We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures which we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.

In this paper, we build on the biquasiles and dual graph diagrams introduced in [7]. We introduce biquasile Boltzmann weights that enhance the previous knot coloring invariant defined in terms of finite biquasiles and provide examples differentiating links with the same counting invariant, demonstrating that the enhancement is proper. We identify conditions for a linear function $\varphi :{\mathbb{Z}}_n[X^3]\to {\mathbb{Z}}_n$ to be a Boltzmann weight for an Alexander biquasile $X$.

We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular, we show that the invariant can distinguish certain virtual knots.