Markov Theorem For Free Links
Vassily Olegovich Manturov, Hang Wang
TL;DR
This paper introduces the concept of free links, defines free braids, and establishes a Markov theorem that characterizes when two free braids produce the same free link, aiding the study of free knots and links.
Contribution
It defines free braids, proves an Alexander theorem for free links, and establishes a Markov theorem for free braids, extending classical link theory to free links.
Findings
All free links can be represented as closures of free braids.
Necessary and sufficient conditions are provided for free braids to have the same closure.
The results facilitate the study of invariants for free knots and links.
Abstract
The notion of free link is a generalized notion of virtual link. In the present paper we define the group of free braids, prove the Alexander theorem that all free links can be obtained as closures of free braids and prove a Markov theorem, which gives necessary and sufficient conditions for two free braids to have the same free link closure. Our result is expected to be useful in study the topology invariants for free knots and links.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Advanced Operator Algebra Research