Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Homfly and dichromatic polynomials


Authors: Xian’an Jin and Fuji Zhang
Journal: Proc. Amer. Math. Soc. 140 (2012), 1459-1472
MSC (2000): Primary 57M15, 57M27; Secondary 05CXX
DOI: https://doi.org/10.1090/S0002-9939-2011-11050-5
Published electronically: August 1, 2011
MathSciNet review: 2869131
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we first associate a plane graph with an oriented link via replacing each edge of the graph by an alternatingly oriented 2-tangle. Then we establish a relation between the Homfly polynomial of the associated oriented link and the dichromatic polynomial of the plane graph by assigning suitable edge weights. This result extends work of F. Jaeger and L. Traldi.


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

  • 1. J. W. Alexander, Topological invariants of knots and links, Tran. Amer. Math. Soc. 30 (1928), 275-306. MR 1501429
  • 2. B. Bollobaś, Modern Graph Theory, Springer, 1998. MR 1633290 (99h:05001)
  • 3. B. Bollobás, O. Riordan, A Tutte polynomial for colored graphs, Combin. Probab. Comput. 8 (1999), 45-93. MR 1684623 (2000f:05033)
  • 4. T. Brylawski, A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971), 1-22. MR 0288039 (44:5237)
  • 5. J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Comput. Problems in Abstract Algebra, Pergamon Press, New York (1970), 329-358. MR 0258014 (41:2661)
  • 6. P. G. Cromwell, Knots and Links, Cambridge University Press, 2004. MR 2107964 (2005k:57011)
  • 7. Y. Diao, C. Ernst, U. Zieglerc, Jones polynomial of knots formed by repeated tangle replacement operations, Toplogy Appl. 156 (2009), 2226-2239. MR 2544132 (2010k:57025)
  • 8. Y. Diao, G. Hetyei, K. Hinson, Tutte polynomials of tensor products of signed graphs and their applications in knot theory, J. Knot Theory Ramification 18 (2009), 561-590. MR 2527677 (2010c:57010)
  • 9. S. Eliahou, L. H. Kauffman, M. B. Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42, no. 1 (2003), 155-169. MR 1928648 (2003g:57015)
  • 10. P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 239-246. MR 776477 (86e:57007)
  • 11. F. Jaeger, Tutte polynomials and link polynomials, Proc. Amer. Math. Soc. 103 (1988), 647-654. MR 943099 (89i:57004)
  • 12. X. Jin, F. Zhang, F. Dong, E. G. Tay, Zeros of the Jones polynomial are dense in the complex plane, Electron. J. Comb. 17, no. 1 (2010) R94. MR 2661397
  • 13. X. Jin, F. Zhang, The Homfly polynomial for even polyhedral links, MATCH Commun. Math. Comput. Chem. 63 (2010), 657-677. MR 2666626 (2011c:92100)
  • 14. V. F. R. Jones, A polynomial invariant for knots via Von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 103-111. MR 766964 (86e:57006)
  • 15. L. H. Kauffman, A Tutte polynomial for signed graphs, Discrete Appl. Math. 25 (1989), 105-127. MR 1031266 (91c:05082)
  • 16. S. Liu, X. Cheng, H. Zhang, W. Qiu, The architecture of polyhedral links and their HOMFLY polynomials, J. Math. Chem. 48, no. 2 (2010), 439-456. MR 2665344
  • 17. K. Luse, Y. Rong, Examples of knots with the same polynomials, J. Knot Theory Ramification 15, no. 6 (2006), 749-759. MR 2253833 (2007f:57019)
  • 18. J. G. Oxley, D. J. A. Welsh, Tutte polynomials computable in polynomial time, Discrete Math. 109 (1992), 185-192. MR 1192381 (94e:05076)
  • 19. J. H. Przytycki, P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987), 115-139. MR 945888 (89h:57006)
  • 20. R. C. Read, E. G. Whitehead Jr., Chromatic polynomials of homeomorphism classes of graphs, Discrete Math. 204 (1999), 337-356. MR 1691877 (2000b:05059)
  • 21. W. Schwärzler, D. J. A. Welsh, Knots, matroids and the Ising model, Math. Proc. Cambridge Philos. Soc. 113, no. 1 (1993), 107-139. MR 1188822 (94c:57019)
  • 22. L. Traldi, A dichromatic polynomial for weighted graphs and link polynomials, Proc. Amer. Math. Soc. 106 (1989), 279-286. MR 955462 (90a:57013)
  • 23. L. Traldi, Parallel connections and colored Tutte polynomials, Discrete Math. 290 (2005), 291-299. MR 2123398 (2005j:05033)
  • 24. W. T. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math. 6 (1954), 80-91. MR 0061366 (15:814c)
  • 25. L. Waston, Any tangle extends to non-mutant knots with the same Jones polynomials, J. Knot Theory Ramification 15, no. 9 (2006), 1153-1162. MR 2287438 (2007k:57028)
  • 26. D. R. Woodall, Tutte polynomial expansions for $ 2$-separable graphs, Discrete Math. 247 (2002), 201-213. MR 1893028 (2002m:05094)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57M15, 57M27, 05CXX

Retrieve articles in all journals with MSC (2000): 57M15, 57M27, 05CXX


Additional Information

Xian’an Jin
Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China
Email: xajin@xmu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2011-11050-5
Keywords: Link, oriented tangle, graph, Homfly polynomial, dichromatic polynomial.
Received by editor(s): December 29, 2010
Published electronically: August 1, 2011
Communicated by: Jim Haglund
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society