The classification of partially symmetric 3-braid links

• Autorzy: Alexander Stoimenov
• Języki publikacji: EN

Abstrakty

EN

We classify 3-braid links which are amphicheiral as unoriented links, including a new proof of Birman- Menasco’s result for the (orientedly) amphicheiral 3-braid links. Then we classify the partially invertible 3-braid links.

Słowa kluczowe

EN

Jones polynomial; 3-braid link; Amphicheira; Invertible; Semiadequate link; Genus

Kategorie tematyczne

• 05C62: Graph representations (geometric and intersection representations, etc.) {For graph drawing, see also }
• 05E15: Combinatorial aspects of groups and algebras
• 20B40: Computational methods
• 57M25: Knots and links in S 3
• 20F36: Braid groups; Artin groups

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

otrzymano

2015-01-22

zaakceptowano

2015-06-25

online

2015-08-11

Twórcy

autor

Alexander Stoimenov

• Gwangju Institute of Science and Technology, School of General Studies,
  GIST College, 123 Cheomdan-gwagiro, Gwangju 500-712, Korea

Bibliografia

• ---
• [1] Bennequin, D. , Entrelacements et équations de Pfaff , Soc. Math. de France, Astérisque 107-108 (1983), 87–161.
• [2] Birman, J. S. , On the Jones polynomial of closed 3-braids, Invent. Math. 1985, 81(2), 287–294.
• [3] -"- , New points of view in knot theory, Bull. Amer. Math. Soc. 1993, 28(2), 253–287.
• [4] -"- and Menasco, W. W. , Studying knots via braids III: Classifying knots which are closed 3 braids, Pacific J. Math. 1993, 161, 25–113.
• [5] Brandt, R. D. , Lickorish, W. B. R. and Millett, K. , A polynomial invariant for unoriented knots and links, Inv. Math. 1986, 74, 563–573.
• [6] Cromwell, P. R. , Homogeneous links, J. London Math. Soc. (series 2) 1989, 39, 535–552.
• [7] Dasbach, O. and Gemein, B. , A Faithful Representation of the Singular Braid Monoid on Three Strands, Proceedings of the International Conference on Knot Theory “Knots in Hellas, 98", Series on Knots and Everything 24, World Scientific, 2000, 48–58.
• [8] -"- and Lin, X.-S. , A volume-ish theorem for the Jones polynomial of alternating knots, Pacific J. Math. 2007, 231(2), 279–291.
• [9] -"- and " , On the Head and the Tail of the Colored Jones Polynomial, Compositio Math. 2006, 142(5), 1332–1342.
• [10] Freyd, P. , Hoste, J. , Lickorish, W. B. R. , Millett, K. , Ocneanu, A. and Yetter, D. , A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 1985, 12, 239–246. [Crossref]
• [11] Futer, D. , Kalfagianni, E. and Purcell, J. , Cusp areas of Farey manifolds and applications to knot theory, Int. Math. Res. Not. 2010, 23, 4434–4497. [WoS]
• [12] Gabai, D. , The Murasugi sum is a natural geometric operation II, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), 93–100, Contemp. Math. 44, Amer. Math. Soc., Providence, RI, 1985.
• [13] -"- , The Murasugi sum is a natural geometric operation, Low-dimensional topology (San Francisco, Calif., 1981), Contemp. Math. 20, 131–143, Amer. Math. Soc., Providence, RI, 1983.
• [14] Hosokawa, F. , On r-polynomials of links, Osaka Math. J. 1958, 10, 273–282.
• [15] Hoste, J. , The first coefficient of the Conway polynomial, Proc. Amer. Math. Soc. 1985, 95(2), 299–302.
• [16] Jones, V. F. R. , Hecke algebra representations of of braid groups and link polynomials, Ann. of Math. 1987, 126, 335–388.
• [17] Kanenobu, T. , Relations between the Jones and Q polynomials of 2-bridge and 3-braid links, Math. Ann. 1989, 285, 115–124.
• [18] Kauffman, L. H. , State models and the Jones polynomial, Topology 1987, 26, 395–407. [Crossref]
• [19] Kirby, R. (ed.), Problems of low-dimensional topology, book available on http://math.berkeley.edu/˜kirby.
• [20] Lickorish, W. B. R. and Millett, K. C. , The reversing result for the Jones polynomial, Pacific J. Math. 1986, 124(1), 173–176.
• [21] -"- and Thistlethwaite, M. B. , Some links with non-trivial polynomials and their crossing numbers, Comment. Math. Helv. 1988, 63, 527–539.
• [22] Menasco, W. W. and Thistlethwaite, M. B. , The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (2) (1991), 403–412. [Crossref]
• [23] Murakami, J. , The Kauffman polynomial of links and representation theory, Osaka J. Math. 1987, 24(4), 745–758.
• [24] Przytycki, J. and Traczyk, P. , Invariants of links of Conway type, Kobe J. Math. 1988, 4(2), 115–139.
• [25] Rolfsen, D. , Knots and links, Publish or Parish, 1976.
• [26] Schreier, O. , Über die Gruppen AaBb D 1, Abh. Math. Sem. Univ. Hamburg 1924, 3, 167–169. [Crossref]
• [27] Stoimenow, A. , The skein polynomial of closed 3-braids, J. Reine Angew. Math. 2003, 564, 167–180.
• [28] -"- , Properties of closed 3-braids, preprint math.GT/0606435.
• [29] -"- , Coefficients and non-triviality of the Jones polynomial, J. Reine Angew. Math. 2011, 657, 1–55. [WoS]
• [30] Thistlethwaite, M. B. , On the Kauffman polynomial of an adequate link, Invent. Math. 93(2) (1988), 285–296. [Crossref]
• [31] -"- , A spanning tree expansion for the Jones polynomial, Topology 1987, 26, 297–309.
• [32] Xu, P. , The genus of closed 3-braids, J. Knot Theory Ramifications 1992, 1(3), 303–326.

Identyfikatory

DOI

10.1515/math-2015-0043