Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
Full-text PDF

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 $ 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.


References [Enhancements On Off] (What's this?)

  • 1. B. Bollobas, Modern Graph Theory, Springer-Verlag, New York, 1998. MR 1633290 (99h:05001)
  • 2. J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, North-Holland, Amsterdam. 1981.
  • 3. A. Brøndsted, An Introduction to Convex Polytopes, Springer-Verlag, New York, 1982. MR 683612 (84d:52009)
  • 4. B.R. Heap, The enumeration of homeomorphically irreducible star graphs, Journal of Mathematical Physics, 7 (1966), 1852-1857. MR 0202638 (34:2500)
  • 5. L.H. Kauffman, A Tutte polynomial for signed graphs, Discrete Appl. Math. 25 (1989), 105-127. MR 1031266 (91c:05082)
  • 6. L.H. Kauffman, Knots and Physics, third edition, World Scientific Publishing Co., River Edge, NJ, 2001. MR 1858113 (2002h:57012)
  • 7. R.C. Read and E.G. Whitehead, Chromatic polynomials of homeomorphism classes of graphs, Discrete Math. 204 (1999), 337-356. MR 1691877 (2000b:05059)
  • 8. Weiling Yang and Fuji Zhang, The Kauffman bracket polynomial of links and universal signed plane graph, Lecture Notes in Computer Science 4381, 228-244, Springer-Verlag, New York, 2007.
  • 9. Weiling Yang and Xian'an Jin, The construction of 2-connected plane graph with cyclomatic number 5, (in Chinese) Journal of Mathematical Study, 37 (2004), 83-95. MR 2229412
  • 10. Xian'an Jin and Fuji Zhang, The Kauffman brackets for equivalence classes of links, Advances in Appl. Math. 34 (2005), 47-64. MR 2102274 (2005j:57009)
  • 11. R.C. Read, Chain polynomials of graphs, Discrete Math. 265 (2003), 213-235. MR 1969375 (2004c:05074)
  • 12. http://math.xmu.edu.cn/school/teacher/fzzhang/fuji_zhang.html

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 05C10, 57M25

Retrieve articles in all journals with MSC (2000): 05C10, 57M25


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.

American Mathematical Society