On invariants for surface-links in entropic magmas via marked graph diagrams

TL;DR

This paper introduces a generalized algebraic structure called marked Kauffman bracket magma to define invariants for surface-links in 4-space, extending previous invariants for classical links.

Contribution

It formulates a new algebraic framework, marked Kauffman bracket magma, enabling invariants for surface-links via marked graph diagrams.

Findings

Defined conditions for invariance under Yoshikawa moves

Constructed invariants for surface-links in 4-space

Reformulated the multiplication using a map from links to magma

Abstract

M. Niebrzydowski and J. H. Przytycki defined a Kauffman bracket magma and constructed the invariant P of framed links in 3-space. The invariant is closely related to the Kauffman bracket polynomial. The normalized bracket polynomial is obtained from the Kauffman bracket polynomial by the multiplication of indeterminate and it is an ambient isotopy invariant for links. In this paper, we reformulate the multiplication by using a map from the set of framed links to a Kauffman bracket magma in order that $P$ is invariant for links in 3-space. We define a generalization of a Kauffman bracket magma, which is called a marked Kauffman bracket magma. We find the conditions to be invariant under Yoshikawa moves except the first one and use a map from the set of admissible marked graph diagrams to a marked Kauffman bracket magma to obtain the invariant for surface-links in 4-space.