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Xu’s Form and Seifert Surfaces

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Abstract

We studied the relation of Xu’s algorithm and the skein polynomial in [St2]***, and here we will go further to connect fiberedness, and the Alexander and Jones polynomial to Xu’s form.

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References

  1. J.S. Birman, On the Jones polynomial of closed 3-braids. Invent. Math. 81(2), 287–294 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. J.S. Birman, W.W. Menasco, Studying knots via braids III: classifying knots which are closed 3 braids. Pac. J. Math. 161, 25–113 (1993)

    Article  MATH  Google Scholar 

  3. J.S. Birman, W.W. Menasco, Studying links via closed braids II: On a theorem of Bennequin. Topol. Appl. 40(1), 71–82 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Gabai, The Murasugi sum is a natural geometric operation, in Low-Dimensional Topology (San Francisco, California, 1981). Contemporary Mathematics, vol. 20 (American Mathematical Society, Providence, RI, 1983), pp. 131–143

    Google Scholar 

  5. D. Gabai, The Murasugi sum is a natural geometric operation II, in Combinatorial Methods in Topology and Algebraic Geometry (Rochester, N.Y., 1982). Contemporary Mathematics, vol. 44 (American Mathematical Society, Providence, RI, 1985), pp. 93–100

    Google Scholar 

  6. D. Gabai, Detecting fibred links in S 3. Comment. Math. Helv. 61(4), 519–555 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Hoste, M. Thistlethwaite, KnotScape, a knot polynomial calculation and table access program. Available at http://www.math.utk.edu/~morwen

  8. J.A. Kneissler, Woven braids and their closures. J. Knot Theory Ramifications 8(2), 201–214 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Kobayashi, Uniqueness of minimal genus Seifert surfaces for links. Topol. Appl. 33(3), 265–279 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. W.B.R. Lickorish, K.C. Millett, A polynomial invariant for oriented links. Topology 26(1), 107–141 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Rudolph, Braided surfaces and Seifert ribbons for closed braids. Comment. Math. Helv. 58, 1–37 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Stoimenow, On polynomials and surfaces of variously positive links. J. Eur. Math. Soc. 7(4), 477–509 (2005). math.GT/0202226

    Google Scholar 

  13. A. Stoimenow, The skein polynomial of closed 3-braids. J. Reine Angew. Math. 564, 167–180 (2003)

    MathSciNet  MATH  Google Scholar 

  14. A. Stoimenow, Coefficients and non-triviality of the Jones polynomial, J. Reine Angew. Math. 657 (2011), 1–55; see also math.GT/0606255

    Google Scholar 

  15. P. Xu, The genus of closed 3-braids. J. Knot Theory Ramifications 1(3), 303–326 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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Stoimenow, A. (2017). Xu’s Form and Seifert Surfaces. In: Properties of Closed 3-Braids and Braid Representations of Links . SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-68149-8_3

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