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A BRACKET POLYNOMIAL FOR GRAPHS, I

LORENZO TRALDI

Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042, USA

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and

LOUIS ZULLI

Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042, USA

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https://doi.org/10.1142/S021821650900766XCited by:18 (Source: Crossref)

Abstract

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial V_G(t) that is invariant under the graph Reidemeister moves.

Keywords:

AMSC: 57M25, 05C50

References

R. Arratia, B. Bollobás and G. B. Sorkin, The interlace polynomial: A new graph polynomial, Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (Association for Computing Machinery, New York, 2000) pp. 237–245. Google Scholar
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
R. Arratia, B. Bollobás and G. B. Sorkin, Combinatorica 24, 567 (2004), DOI: 10.1007/s00493-004-0035-6. Crossref, Web of Science, Google Scholar
P. N. Balisteret al., Europ. J. Combin. 22, 1 (2001), DOI: 10.1006/eujc.2000.0434. Crossref, Web of Science, Google Scholar
J. S. Birman and T. Kanenobu, Proc. Amer. Math. Soc. 102, 687 (1988), DOI: 10.2307/2047247. Web of Science, Google Scholar
A. Bouchet, J. Combin. Theory Ser. B 60, 107 (1994), DOI: 10.1006/jctb.1994.1008. Crossref, Web of Science, Google Scholar
A. Bouchet, Europ. J. Combin. 22, 657 (2001), DOI: 10.1006/eujc.2000.0486. Crossref, Web of Science, Google Scholar
S. V. Chmutov and S. K. Lando, Algebr. Geom. Topol. 7, 1579 (2007), DOI: 10.2140/agt.2007.7.1579. Crossref, Web of Science, Google Scholar
C. F. Gauss , Werke, Band VIII ( Teubner , Leipzig , 1900 ) . Google Scholar
S. Huggett, J. Knot Theory Ramifications 14, 919 (2005), DOI: 10.1142/S0218216505004147. Link, Web of Science, Google Scholar
F. Jaeger, D. L. Vertigan and D. J. A. Welsh, Math. Proc. Cambridge Philos. Soc. 108, 35 (1990), DOI: 10.1017/S0305004100068936. Crossref, Web of Science, 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
L. H. Kauffman, Atti Semin. Mat. Fis. Univ. Modena 49, 241 (2001). Google Scholar
L. H. Kauffman, Handbook of Knot Theory, eds. W. Menasco and M. Thistlethwaite (Elsevier B. V., Amsterdam, 2005) pp. 233–318. Crossref, Google Scholar
L. H. Kauffman, Topology 26, 395 (1987). 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
J. Lannes, Comment. Math. Helv. 60, 179 (1985), DOI: 10.1007/BF02567409. Crossref, Web of Science, Google Scholar
O.-P. Östlund, J. Knot Theory Ramifications 10, 1215 (2001). Link, Web of Science, Google Scholar
K. Reidemeister , Knotentheorie ( Springer , Berlin , 1932 ) . Google Scholar
B. Shtylla and L. Zulli, J. Knot Theory Ramifications 15, 81 (2006), DOI: 10.1142/S0218216506004294. 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. Zulli, Topology 34, 717 (1995), DOI: 10.1016/0040-9383(94)00041-I. Crossref, Web of Science, Google Scholar