Links and cubic 3-polytopes

Yang, Weiling; Zhang, Fuji

doi:10.1090/S0025-5718-08-02088-7

Links and cubic 3-polytopes
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by Weiling Yang and Fuji Zhang PDF

Math. Comp. 77 (2008), 1841-1857 Request permission

Abstract:

It is well known that a prime link diagram corresponds to a signed plane graph without cut vertices (Kauffman, 1989). In this paper, we present a new relation between prime links and cubic 3-polytopes. Let $S$ be the set of links such that each $L\in S$ has a diagram whose corresponding signed plane graph is the graph of a cubic 3-polytope. We show that all nontrivial prime links, except $(2,n)$-torus links and $(p,q,r)$-pretzel links, can be obtained from $S$ by using some operation of untwining. Furthermore, we define the generalized cubic 3-polytope chains and then show that any nontrivial link can be obtained from $\mathbb {S}$ by some untwining operations, where $\mathbb {S}$ is the set of links corresponding to generalized cubic 3-polytope chains. These results are used to simplify the computation of the Kauffman brackets of links so that the computing can be done in a unified way for many infinite families of links.

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