Narrow Results

Ito–Takimura recently defined a splice-unknotting number $u^{-}(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.

This paper studies the biquandle as an invariant of virtual knots and links, analyzing its structure via composition of biquandle morphisms where elementary morphisms are associated with the crossings and virtual crossings in the diagram.

We introduce the even index polynomial of virtual link diagrams and some even state sum polynomials which are the even Jones–Kauffman, even Miyazawa and even arrow polynomials. By simple computation, we can distinguish virtual links with these even polynomial invariants. Also, we give examples so that these invariants are different from the original ones.

The problem concerning which Gauss diagrams can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows “the number of exits = the number of entrances” and the sufficient condition is based on the Jordan curve theorem. Further, using matrices, we redefine conditions for realizability of Gauss diagrams and then we give an algorithm to construct meanders.

Twisted knot theory, introduced by Bourgoin, is a generalization of virtual knot theory. It is well known that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two forbidden moves $F1$ and $F2$. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three forbidden moves $T4$, $F1$ (or $F2$) and $F3$ (or $F4$).