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New presentations of a link and virtual link

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China

https://doi.org/10.1142/S0218216519500512Cited by:0 (Source: Crossref)

Abstract

An embedding presentation of a diagram is introduced, which has proved to be a unique presentation of a diagram. Let $ℒ$ be a set of all diagrams, called also links in this paper. An algebraic system $(ℒ, \sim)$ is constructed. In fact, a link in $R^{3}$ (or $S^{3}$ ) is the equivalent class $[L]$ where $L$ is one of its embedding presentations. Based on $(ℒ, \sim)$ , Reduction Crossing Algorithm is proposed which is used to reduce the number of crossings in an embedding presentation by introducing a main tool called a pass replacement. For an infinite set of unknots $𝒰$ , each $K$ in $𝒰$ can be transformed into the trivial unknot in at most $O (n^{c})$ by applying the algorithm where $c$ is a constant, $K \in 𝒰$ and $n = | V (K) |$ . As special consequences, three unknots are unknotted, which are Goeritz’s unknot, Thistlethwaite’s unknot and Haken’s unknot (image courtesy of Cameron Gordon). Moreover, an infinite family of unknots $K_{G_{2 k, 2 l}} \in 𝒰$ are unknotted in $O (n log log n)$ time. In addition, unique presentations of a virtual link, an oriented link and oriented virtual link are introduced, respectively.

Keywords:

AMSC: 57M25

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