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Generalizations of a Conway algebra for oriented surface-links via marked graph diagrams. (English) Zbl 1406.57004

A Conway algebra is an algebraic structure which defines a HOMFLY-PT polynomial for a knot, introduced by J. H. Przytycki and P. Traczyk [Kobe J. Math. 4, No. 2, 115–139 (1987; Zbl 0655.57002)]. In this paper, the authors gave an invariant for an oriented link in \(\mathbb{R}^3\), valued in a Conway algebra.
In the paper under review, the main results are as follow. The authors introduce the notion of a marked Conway algebra, which is a generalization of Conway algebras. They define an invariant for oriented marked graphs, valued in the marked Conway algebra. By considering the marked Conway algebra with certain additional conditions, this invariant becomes an invariant for oriented surface-links. The authors call this invariant the marked Conway type invariant, and this invariant is closely related to the polynomial invariant due to Y. Joung et al. [Topology Appl. 231, 159–185 (2017; Zbl 1379.57032)]. Further, the authors generalize a marked Conway algebra and construct an invariant for oriented marked graphs, valued in a generalized marked Conway algebra. This invariant is defined by applying two skein relations to self and mixed crossings. The idea of this generalization of a marked Conway algebra is similar to the one given by the third author on a generalization of a Conway algebra [S. Kim, J. Knot Theory Ramifications 27, No. 2, Article ID 1850014, 20 p. (2018; Zbl 1385.57016)].
We review the main terms. A surface-link is the image of a smooth embedding of a closed surface in \(\mathbb{R}^4\). A marked graph is a 4-valent graph embedded in \(\mathbb{R}^3\) such that each 4-valent vertex is equipped with a marker. For a marked graph, its diagram in \(\mathbb{R}^2\) is called a marked graph diagram. An oriented marked graph (respectively, an oriented surface-link) is an equivalence class of oriented marked graph diagrams modulo local moves called oriented Yoshikawa moves of type 1 (respectively, oriented Yoshikawa moves of types 1 and 2). The authors consider invariants for oriented marked graph diagrams under moves of these two cases.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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References:

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