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A BRACKET POLYNOMIAL FOR GRAPHS, III: VERTEX WEIGHTS

LORENZO TRALDI

Lafayette College, Easton, Pennsylvania 18042, USA

https://doi.org/10.1142/S0218216511008875Cited by:2 (Source: Crossref)

Abstract

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

Keywords:

AMSC: 57M25, 05C50

References

R. Arratia, B. Bollobás and G. B. Sorkin, J. Combin. Theory Ser. B 92, 199 (2004), DOI: 10.1016/j.jctb.2004.03.003. Crossref, Web of Science, Google Scholar
P. N. Balisteret al., Europ. J. Combin. 23, 761 (2002), DOI: 10.1006/eujc.2002.0602. Crossref, Web of Science, Google Scholar
I. Beck, J. Combin. Theory Ser. A 23, 198 (1977). Crossref, Web of Science, Google Scholar
B. Bollobás and O. Riordan, Proc. London Math. Soc. 83, 513 (2001). Crossref, Web of Science, Google Scholar
B. Bollobás and O. Riordan, Math. Ann. 323, 81 (2002). Crossref, Web of Science, Google Scholar
H. R. Brahana, Ann. Math. 23, 144 (1921), DOI: 10.2307/1968030. Crossref, Google Scholar
T. Brylawski, Trans. Amer. Math. Soc. 154, 1 (1971), DOI: 10.1090/S0002-9947-1971-0288039-7. Crossref, Web of Science, Google Scholar
S. Chaiken, J. Combin. Theory Ser. B 46, 96 (1989). Crossref, Web of Science, Google Scholar
S. Chmutov and I. Pak, Moscow Math. J. 7, 409 (2007). Crossref, Web of Science, Google Scholar
S. Chmutov and J. Voltz, J. Knot Theory Ramifications 17, 1189 (2008). Link, Web of Science, Google Scholar
M. Cohn and A. Lempel, J. Combin. Theory Ser. A 13, 83 (1972). Crossref, Web of Science, Google Scholar
J. H. Conway, Computational Problems in Abstract Algebra (Pergamon, Oxford, UK, 1970) pp. 329–358. Crossref, Google Scholar
W. H. Cunningham, SIAM J. Alg. Disc. Meth. 3, 214 (1982), DOI: 10.1137/0603021. Crossref, Web of Science, Google Scholar
Y. Diao, C. Ernst and U. Ziegler, Topol. Appl. 156, 2226 (2009), DOI: 10.1016/j.topol.2009.05.006. Crossref, Web of Science, Google Scholar
Y. Diao and G. Hetyei, Combin. Probab. Comput. 19, 343 (2010). Crossref, Web of Science, Google Scholar
J. A. Ellis-Monaghan and I. Sarmiento, Congr. Numer. 155, 57 (2002). Google Scholar
D. P. Ilyutko and V. O. Manturov, J. Knot Theory Ramifications 18, 791 (2009). Link, Web of Science, Google Scholar
F. Jaeger, Cycles and Rays, NATO Advanced Science Insitutes Series C: Mathematical and Physical Sciences 301, ed. P. Q. Montreal (Kluwer, Dordrecht, 1987) pp. 123–150. Google Scholar
V. F. R. Jones, Bull. Amer. Math. Soc. 12, 103 (1985), DOI: 10.1090/S0273-0979-1985-15304-2. Crossref, Web of Science, Google Scholar
N. Kamada, Osaka J. Math. 39, 325 (2002). Web of Science, Google Scholar
L. H. Kauffman, Topology 26, 395 (1987). Crossref, Web of Science, Google Scholar
L. H. Kauffman, Discrete Appl. Math. 25, 105 (1989), DOI: 10.1016/0166-218X(89)90049-8. Crossref, Web of Science, Google Scholar
L. H. Kauffman, Europ. J. Combin. 20, 663 (1999), DOI: 10.1006/eujc.1999.0314. Crossref, Web of Science, Google Scholar
L. H. Kauffman , Handbook of Knot Theory , eds. W. Menasco and M. Thistlethwaite ( Elsevier , Amsterdam , 2005 ) . Crossref, Google Scholar
A. Kotzig, Theory of Graphs (Academic Press, New York, 1968) pp. 219–230. Google Scholar
P. Martin, Enumérations eulériennes dans les multigraphes et invariants de Tutte-Grothendieck, Thèse, Grenoble (1977) . Google Scholar
B. Mellor, Michigan Math. J. 51, 509 (2003), DOI: 10.1307/mmj/1070919557. Crossref, Web of Science, Google Scholar
K. Murasugi, Trans. Amer. Math. Soc. 314, 1 (1989), DOI: 10.1090/S0002-9947-1989-0930077-6. Crossref, Web of Science, Google Scholar
R. C. Read and P. Rosenstiehl, Combinatorics, Colloq. Math. Soc. János Bolyai 18 (North-Holland, Amsterdam-New York, 1978) pp. 843–876. Google Scholar
N. D. A. Ryder, Skein based invariants and the Kauffman polynomial, Ph.D. dissertation, University of Liverpool (2008) . Google Scholar
W. Schwärzler and D. J. A. Welsh, Math. Proc. Cambridge Phil. Soc. 115, 107 (1993). Google Scholar
E. Soboleva, J. Knot Theory Ramifications 10, 161 (2001). Link, Web of Science, Google Scholar
M. B. Thistlethwaite, Topology 26, 297 (1987), DOI: 10.1016/0040-9383(87)90003-6. Crossref, Web of Science, Google Scholar
L. Traldi, Proc. Amer. Math. Soc. 106, 279 (1989), DOI: 10.1090/S0002-9939-1989-0955462-3. Crossref, Web of Science, Google Scholar
L. Traldi, Binary nullity, Euler circuits and interlace polynomials, to appear in Europ. J. Combin . Google Scholar
L. Traldi, Combin. Probab. Comput. 19, 133 (2010). Crossref, Web of Science, Google Scholar
L. Traldi, J. Knot Theory Ramifications 19(4), 547 (2010). Link, Web of Science, Google Scholar
L. Traldi and L. Zulli, J. Knot Theory Ramifications 18(12), 1681 (2009). Link, Web of Science, Google Scholar
L. Zulli, Topology 34, 717 (1995), DOI: 10.1016/0040-9383(94)00041-I. Crossref, Web of Science, Google Scholar