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GRAPHICAL CALCULI FOR THE DUBROVNIK POLYNOMIAL WITH APPLICATIONS

XIAN'AN JIN

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, P. R. China

https://doi.org/10.1142/S0218216512501003Cited by:0 (Source: Crossref)

Abstract

We introduce a polynomial for plane graphs, which is proposed to equal the Dubrovnik polynomial of the corresponding alternating link diagrams via the medial construction. Then using this polynomial we define another polynomial for doubly edge-weighted plane graphs, which has a natural connection with Dubrovnik polynomial of links formed from plane graphs by edge-tangle replacements. In the final, via this connection we give an explanation for a result due to Lipson and compute the Dubrovnik polynomials of classical pretzel links.

Keywords:

AMSC: 57M25, 57M15

References

M. Aigner, Math. Ann. 307, 173 (1997). Crossref, Web of Science, Google Scholar
J. A. Bondy and U. S. R. Murty , Graph Theory with Applications ( Macmillan Press Ltd , 1976 ) . Crossref, Google Scholar
R. P. Carpentier, J. Knot Theory Ramifications 9, 975 (2000). Link, Web of Science, Google Scholar
Y. Diao, G. Hetyei and K. Hinson, J. Knot Theory Ramifications 18, 561 (2009). Link, Web of Science, Google Scholar
F. Jaeger, Proc. Amer. Math. Soc. 103, 647 (1988). Crossref, Web of Science, Google Scholar
F. Jaeger, On the Kauffman polynomial of planar matroids, Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (1992) pp. 117–127. Google Scholar
F. Jaeger, D. L. Vertigan and D. J. A. Welsh, Math. Proc. Cambridge Philos. Soc. 108, 35 (1990). Crossref, Web of Science, Google Scholar
X. Jin and F. Zhang, Adv. Appl. Math. 34, 47 (2005). Crossref, Web of Science, Google Scholar
X. Jin and F. Zhang, Adv. Appl. Math. 39, 155 (2007). Crossref, Web of Science, Google Scholar
X. Jinet al., Electron. J. Combin. 17(1), R94 (2010). Crossref, Google Scholar
X. Jin and F. Zhang, Proc. Amer. Math. Soc. 140, 1459 (2012). Crossref, Web of Science, Google Scholar
V. F. R. Jones, Bull. Amer. Math. Soc. 12, 103 (1985). Crossref, Web of Science, Google Scholar
L. H. Kauffman, Topology 26, 395 (1987). Crossref, Web of Science, Google Scholar
L. H. Kauffman, Trans. Amer. Math. Soc. 311(2), 697 (1989). Crossref, Web of Science, Google Scholar
L. H. Kauffman, Trans. Amer. Math. Soc. 318(2), 417 (1990). Crossref, Web of Science, Google Scholar
L. H. Kauffman and P. Vogel, J. Knot Theory Ramifications 1, 59 (1992). Link, Google Scholar
M. E. Kidwell, Proc. Amer. Math. Soc. 100(4), 755 (1987). Crossref, Web of Science, Google Scholar
W. B. R. Lickorish, Math. Proc. Phil. Soc. 105, 103 (1989). Crossref, Web of Science, Google Scholar
A. S. Lipson, Enseign. Math. 36, 93 (1990). Web of Science, Google Scholar
K. Murasugi , Knot Theory and its Applications ( Birkhäuser , 1996 ) . Google Scholar
S. D. Noble and D. J. A. Welsh, J. Graph Theory 34, 100 (2000). Crossref, Web of Science, Google Scholar
R. Penrose , Applications of negative dimensional tensors , Combinatorial Mathematics and its Applications: Proceedings of a Conference Held at the Mathematical Institute , ed. D. J. A. Welsh ( Academic Press , London, New York , 1971 ) . Google Scholar
R. C. Read and E. G. Whitehead Jr., Discrete Math. 243, 267 (2002). Crossref, Web of Science, Google Scholar
N. D. A. Ryder, Skein based invariants and the Kauffman polynomial, PhD Thesis, University of Livepool (2008) . Google Scholar
M. B. Thistlethwaite, Topology 26, 297 (1987). Crossref, Web of Science, Google Scholar
L. Traldi, Proc. Amer. Math. Soc. 106, 279 (1989). Crossref, Web of Science, Google Scholar
W. T. Tutte, Canad. J. Math. 6, 80 (1954). Crossref, Web of Science, Google Scholar
D. R. Woodall, Discrete Math. 247, 201 (2002). Crossref, Web of Science, Google Scholar