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Generalizations of a Conway algebra for oriented surface-links via marked graph diagrams

Yongju Bae

Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu, Korea

E-mail Address: ybae@knu.ac.kr

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Seonmi Choi

Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu, Korea

E-mail Address: csm123c@gmail.com

Corresponding author.

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, and

Seongjeong Kim

Department of Fundamental Sciences, Bauman Moscow State Technical University, Moscow, Russia

E-mail Address: ksj19891120@gmail.com

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https://doi.org/10.1142/S0218216518420142Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue — The Fourth Russian-Chinese Conference on Knot Theory and Related Topics; Guest Editors: Z. Cheng, D. A. Fedoseev, L. H. Kauffman, V. O. Manturov, A. Y. Vesnin and Z. Yang

Abstract

In 1987, Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the HOMFLY-PT polynomial invariant. In 2018, Kim generalized a Conway algebra, which is an algebraic structure with two skein relations, which is called a generalized Conway algebra. In 2017, Joung, Kamada, Kawauchi and Lee constructed a polynomial invariant of oriented surface-links by using marked graph diagrams.

In this paper, we will introduce generalizations $M A$ $M A$ and $\hat{M A}$ $\hat{M A}$ of a Conway algebra and a generalized Conway algebra, which are called a marked Conway algebra and a generalized marked Conway algebra, respectively. We will construct invariants valued in $M A$ $M A$ and $\hat{M A}$ $\hat{M A}$ for oriented marked graphs and oriented surface-links by applying binary operations to classical crossings and marked vertices via marked graph diagrams. The polynomial invariant of oriented surface-links is obtained from the invariant valued in the marked Conway algebra with additional conditions.

Keywords:

AMSC: 57M25, 57M27, 57Q45

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