Unknotting twisted knots with Gauss diagram forbidden moves

TL;DR

This paper extends twisted knot theory by demonstrating that twisted knots can be simplified to trivial forms using a finite sequence of extended Reidemeister moves and specific forbidden moves, generalizing prior virtual knot results.

Contribution

It introduces new forbidden moves in twisted knot theory and proves that these moves suffice to unknot any twisted knot, expanding the understanding of knot simplification methods.

Findings

Twisted knots can be deformed into trivial knots using extended moves and forbidden moves.

The paper generalizes virtual knot unknotting results to twisted knots.

New forbidden moves are identified that facilitate unknotting twisted knots.

Abstract

Twisted knot theory, introduced by M.O.Bourgoin, is a generalization of virtual knot theory. It is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two "forbidden moves" $F_{1}$ and $F_{2}$. 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" $T_{4}$, $F_{1}$ (or $F_{2}$) and $F_{3}$ (or $F_{4}$) .