Abstract
Niebrzydowski and Przytycki defined a Kauffman bracket magma and constructed the invariant [Formula: see text] of framed links in [Formula: see text]-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 [Formula: see text] is invariant for links in [Formula: see text]-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 [Formula: see text]-space.
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