Biquasile colorings of oriented surface-links. (English) Zbl 1383.57031
A biquasile is a set with six binary operations satisfying certain axioms, introduced by D. Needell and the second author [J. Knot Theory Ramifications 26, No. 8, Article ID 1750048, 18 p. (2017; Zbl 1372.57030)]. In this paper, by giving biquasile colorings of oriented marked graph diagrams, the authors give the following results. They introduce biquasile colorings of oriented surface-links and define counting invariants for oriented surface-links, which are invariants valued in the non-negative integers. They extend the biquasile Boltzmann weight enhancements to the case of oriented surface-links, from which we can recover the counting invariants. In particular, every Boltzmann enhancement is trivial for closed surface-links. They give several examples.
We review basics. A surface-link is the image of a smooth embedding of a closed surface in \(\mathbb{R}^4\). A marked graph diagram is a planar diagram of a 4-regular spatial graph with a marker at each 4-valent vertex. An “admissible” marked graph diagram represents a closed surface-link, while a non-admissible marked graph diagram represents a cobordism between two classical links. Two oriented marked graph diagrams represent ambient isotopic oriented surface-links if and only if they are related by local moves called Yoshikawa moves besides Reidemeister moves.
We review basics. A surface-link is the image of a smooth embedding of a closed surface in \(\mathbb{R}^4\). A marked graph diagram is a planar diagram of a 4-regular spatial graph with a marker at each 4-valent vertex. An “admissible” marked graph diagram represents a closed surface-link, while a non-admissible marked graph diagram represents a cobordism between two classical links. Two oriented marked graph diagrams represent ambient isotopic oriented surface-links if and only if they are related by local moves called Yoshikawa moves besides Reidemeister moves.
Reviewer: Inasa Nakamura (Kanazawa)
MSC:
| 57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
Citations:
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