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Links and cubic 3-polytopes

Authors: Weiling Yang and Fuji Zhang
Journal: Math. Comp. 77 (2008), 1841-1857
MSC (2000): Primary 05C10; Secondary 57M25
Published electronically: February 14, 2008
MathSciNet review: 2398798
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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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Additional Information

Weiling Yang
Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China

Fuji Zhang
Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China

Keywords: Link diagram, signed plane graph, cubic 3-polytope, generalized cubic 3-polytope chain, untwining, chain polynomial, Kauffman bracket polynomial
Received by editor(s): October 23, 2006
Received by editor(s) in revised form: July 13, 2007
Published electronically: February 14, 2008
Additional Notes: The first author was supported in part by NSFC grant 10501038
The second and corresponding author was supported in part by NSFC grant 10671162
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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