Geodesic
Everywhere Equivalent 3-Braids
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A knot (or link) diagram is said to be everywhere equivalent if all the diagrams obtained by switching one crossing represent the same knot (or link). We classify such diagrams of a closed 3-braid.
Keywords: 3-braid group; Jones polynomial; Kauffman bracket; Burau representation; adequate diagram. 
@article{SIGMA_2014_10_a104,
     author = {Alexander Stoimenow},
     title = {Everywhere {Equivalent} {3-Braids}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a104/}
}
TY  - JOUR
AU  - Alexander Stoimenow
TI  - Everywhere Equivalent 3-Braids
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2014
VL  - 10
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a104/
LA  - en
ID  - SIGMA_2014_10_a104
ER  - 
%0 Journal Article
%A Alexander Stoimenow
%T Everywhere Equivalent 3-Braids
%J Symmetry, integrability and geometry: methods and applications
%D 2014
%V 10
%U http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a104/
%G en
%F SIGMA_2014_10_a104
Alexander Stoimenow. Everywhere Equivalent 3-Braids. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a104/

[1] Bae Y., Morton H. R., “The spread and extreme terms of {J}ones polynomials”, J. Knot Theory Ramifications, 12 (2003), 359–373, arXiv: math.GT/0012089 | DOI | MR | Zbl

[2] Birman J. S., Menasco W. W., “Studying links via closed braids. {III}: {C}lassifying links which are closed {$3$}-braids”, Pacific J. Math., 161 (1993), 25–113 | DOI | MR | Zbl

[3] Cromwell P. R., Knots and links, Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[4] Fiedler T., “On the degree of the {J}ones polynomial”, Topology, 30 (1991), 1–8 | DOI | MR | Zbl

[5] Franks J., Williams R. F., “Braids and the {J}ones polynomial”, Trans. Amer. Math. Soc., 303 (1987), 97–108 | DOI | MR | Zbl

[6] Garside F. A., “The braid group and other groups”, Quart. J. Math. Oxford, 20 (1969), 235–254 | DOI | MR | Zbl

[7] Jones V. F. R., “Hecke algebra representations of braid groups and link polynomials”, Ann. of Math., 126 (1987), 335–388 | DOI | MR | Zbl

[8] Kauffman L. H., “State models and the {J}ones polynomial”, Topology, 26 (1987), 395–407 | DOI | MR | Zbl

[9] Lickorish W. B. R., Millett K. C., “The reversing result for the {J}ones polynomial”, Pacific J. Math., 124 (1986), 173–176 | DOI | MR | Zbl

[10] Lickorish W. B. R., Millett K. C., “A polynomial invariant of oriented links”, Topology, 26 (1987), 107–141 | DOI | MR | Zbl

[11] Lickorish W. B. R., Thistlethwaite M. B., “Some links with nontrivial polynomials and their crossing-numbers”, Comment. Math. Helv., 63 (1988), 527–539 | DOI | MR | Zbl

[12] Morton H. R., “Seifert circles and knot polynomials”, Math. Proc. Cambridge Philos. Soc., 99 (1986), 107–109 | DOI | MR | Zbl

[13] Murasugi K., On closed {$3$}-braids, Memoirs of the American Mathmatical Society, 151, Amer. Math. Soc., Providence, R.I., 1974 | DOI | MR | Zbl

[14] Rolfsen D., Knots and links, Publish or Perish, Berkeley, Calif., 1976 | MR | Zbl

[15] Stoimenow A., “The skein polynomial of closed 3-braids”, J. Reine Angew. Math., 564 (2003), 167–180, arXiv: math.GT/0103041 | DOI | MR

[16] Stoimenow A., “On unknotting numbers and knot trivadjacency”, Math. Scand., 94 (2004), 227–248 | MR | Zbl

[17] Stoimenow A., Properties of closed 3-braids, arXiv: math.GT/0606435

[18] Stoimenow A., “Coefficients and non-triviality of the {J}ones polynomial”, J. Reine Angew. Math., 657 (2011), 1–55 | DOI | MR | Zbl

[19] Stoimenow A., “Everywhere equivalent and everywhere different knot diagrams”, Asian J. Math., 17 (2013), 95–137 | DOI | MR

[20] Stoimenow A., “On the crossing number of semiadequate links”, Forum Math., 26 (2014), 1187–1246 | DOI | MR | Zbl

[21] Stoimenow A., Everywhere equivalent 2-component links, {P}reprint

[22] Thistlethwaite M. B., “On the {K}auffman polynomial of an adequate link”, Invent. Math., 93 (1988), 285–296 | DOI | MR | Zbl

[23] Wolfram S., Mathematica — a system for doing mathematics by computer, Addison-Wesley, Reading, MA, 1988 | Zbl