Narrow Results

In this paper, we generalize virtual knot theory to multi-virtual knot theory where there are a multiplicity of virtual crossings. Each virtual crossing type can detour over the other virtual crossing types, and over classical or immersed crossings. New invariants of multi-virtual knots and links are introduced and new problems that arise are described. We show how the extensions of the Penrose coloring evaluation for trivalent plane graphs and our generalizations of this to non-planar graphs and arbitrary numbers of colors acts as a motivation for the construction of the multi-virtual theory.

As an extension of Reshetikhin and Turaev’s invariant, Costantino, Geer and Patureau-Mirand constructed $3$-manifold invariants in the setting of relative $G$-modular categories, which include both semi-simple and non-semi-simple ribbon tensor categories as examples. In this paper, we follow their method to construct a $3$-manifold invariant from Viro’s $𝔤𝔩(1|1)$-Alexander polynomial. We take lens spaces $L(7,1)$ and $L(7,2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds.

We use quantum invariants to define a 3-manifold invariant j_p which lies in the non-negative integers. We relate j_p to the Heegaard genus, and the cut number. We show that j_p is an invariant of weak p-congruence.