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Unknotting twisted knots with Gauss diagram forbidden moves

Shudan Xue

Center for Combinatorics, Nankai University, Tianjin, Hebei 300071, P. R. China

E-mail Address: 201921001184@smail.xtu.edu.cn

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and

Qingying Deng

School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, P. R. China

E-mail Address: qingying@xtu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0218216523500086Cited by:2 (Source: Crossref)

Abstract

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 $F 1$ $F 1$ and $F 2$ $F 2$ . 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 $T 4$ $T 4$ , $F 1$ $F 1$ (or $F 2$ $F 2$ ) and $F 3$ $F 3$ (or $F 4$ $F 4$ ).

Keywords:

AMSC: 57K12

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