The bracket polynomial of the Celtic link shadow $CK_4^{2n}$

TL;DR

This paper derives the Kauffman bracket polynomial for the Celtic link shadow $CK_4^{2n}$ using recursive and algebraic methods, providing explicit formulas for this class of links.

Contribution

It introduces two complementary approaches—recursive diagrammatic relations and 4-tangle algebraic framework—to compute the bracket polynomial of Celtic link shadows.

Findings

Derived explicit recursive formulas for the bracket polynomial.

Developed an algebraic method using 4-tangle concatenation.

Validated the approaches with concrete computations.

Abstract

We derive the Kauffman bracket polynomial for the shadow of the Celtic link $CK_4^{2n}$ using two complementary approaches. The first approach uses a recursive relation within the Celtic framework of Gross and Tucker, based on diagrammatic identities. The second approach makes use of a 4-tangle algebraic framework: a fundamental tangle is concatenated with itself n times to form an iterated composite tangle, and the Kauffman bracket polynomial is computed by decomposing the state space with respect to the basis elements of the 4-strand diagram monoid.