The Kauffman Bracket Expansion of a Generalized Crossing
Keywords:
generalized half-twist, Jones polynomial, Kauffman bracketAbstract
We examine the Kauffman bracket expansion of the generalized crossing $\Delta_n$, a half-twist on $n$ parallel strands, as an element of the Temperley-Lieb algebra with coefficients in $\mathbb{Z}[A,A^{-1}]$. In particular, we determine the minimum and maximum degrees of all possible coefficients appearing in this expansion. Our main theorem shows that the maximum such degree is quadratic in $n$, while the minimum such degree is linear. We also include an appendix with explicit expansions for $n$ at most six.
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