Geodesic
On the combinatorics of smoothing
Contemporary Mathematics. Fundamental Directions, Topology, Tome 51 (2013), pp. 87-109.

Voir la notice de l'article provenant de la source Math-Net.Ru

Many invariants of knots rely upon smoothing the knot at its crossings. To compute them, it is necessary to know how to count the number of connected components the knot diagram is broken into after the smoothing. In this paper, it is shown how to use a modification of a theorem of Zulli together with a modification of the spectral theory of graphs to approach such problems systematically. We give an application to counting subdiagrams of pretzel knots which have one component after oriented and unoriented smoothings. 
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     author = {M. W. Chrisman},
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     url = {http://geodesic.mathdoc.fr/item/CMFD_2013_51_a5/}
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M. W. Chrisman. On the combinatorics of smoothing. Contemporary Mathematics. Fundamental Directions, Topology, Tome 51 (2013), pp. 87-109. http://geodesic.mathdoc.fr/item/CMFD_2013_51_a5/

[1] Brandenbursky M., Polyak M., Link invariants via counting surfaces, Preprint, 2011 | MR

[2] Burde G., Zieschang H., Knots, Walter de Gruyter Co., Berlin, 2003 | MR | Zbl

[3] Chmutov S., Khoury M. C., Rossi A., Polyak–Viro formulas for coefficients of the Conway polynomial, 2008, arXiv: 0810.3146v1[math.GT] | MR

[4] Chmutov S., Polyak M., Elementary combinatorics for the HOMFLYPT polynomial, 2009, arXiv: 0810.4105v2[math.GT] | MR

[5] Chrisman M. W., “Twist lattices and the Jones–Kauffman polynomial for long virtual knots”, J. Knot Theory Ramifications, 19:5 (2010), 655–675 | DOI | MR | Zbl

[6] Chrisman M. W., A lattice of finite-type invariants of virtual knots, In preparation, 2011

[7] Cvetković D., Rowlinson P., Simić S., Eigenspaces of graphs, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[8] Cvetković D., Rowlinson P., Simić S., An introduction to the theory of graph spectra, Cambridge Univ. Press, Cambridge, 2010 | MR | Zbl

[9] Ilyutko D. P., Manturov V. O., Graph-links, arXiv: 1001.0384v1[math.GT] | MR

[10] Ilyutko D., Nikonov I., Manturov V. O., “Parity in knot theory and graph-links”, J. Math. Sci., 193:6 (2013), 809–965 | DOI | MR

[11] Kauffman L. H., “Virtual knot theory”, Eur. J. Combin., 20:7 (1999), 663–690 | DOI | MR | Zbl

[12] Maunder C. R. F., Algebraic topology, Dover Publ. Inc., New York, 1996 | MR

[13] Mostovoy J., Chmutov S., Dushin S., CDBook: Introduction to Vassiliev knot invariants, http://www.math.ohio-state.edu/~chmutov/preprints/

[14] Traldi L., “A bracket polynomial for graphs. II. Links, Euler circuits and marked graphs”, J. Knot Theory Ramifications, 19:4 (2010), 547–586 | DOI | MR | Zbl

[15] Traldi L., Zulli L., “A bracket polynomial for graphs. I”, J. Knot Theory Ramifications, 18:12 (2009), 1681–1709 | DOI | MR | Zbl

[16] Zulli L., “A matrix for computing the Jones polynomial of a knot”, Topology, 34:3 (1995), 717–729 | DOI | MR | Zbl