Links and cubic 3-polytopes

Authors:
Weiling Yang and Fuji Zhang

Journal:
Math. Comp. **77** (2008), 1841-1857

MSC (2000):
Primary 05C10; Secondary 57M25

DOI:
https://doi.org/10.1090/S0025-5718-08-02088-7

Published electronically:
February 14, 2008

MathSciNet review:
2398798

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Abstract | References | Similar Articles | Additional Information

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 be the set of links such that each has a diagram whose corresponding signed plane graph is the graph of a cubic 3-polytope. We show that all nontrivial prime links, except -torus links and -pretzel links, can be obtained from 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 by some untwining operations, where 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

Email:
ywlxmu@163.com

**Fuji Zhang**

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

Email:
fjzhang@xmu.edu.cn

DOI:
https://doi.org/10.1090/S0025-5718-08-02088-7

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.