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PRIMITIVE/SEIFERT KNOTS WHICH ARE NOT TWISTED TORUS KNOT POSITION

  • Received : 2013.11.05
  • Accepted : 2013.11.19
  • Published : 2013.12.25

Abstract

The twisted torus knots and the primitive/Seifert knots both lie on a genus 2 Heegaard surface of $S^3$. In [5], J. Dean used the twisted torus knots to provide an abundance of examples of primitive/Seifert knots. Also he showed that not all twisted torus knots are primitive/Seifert knots. In this paper, we study the other inclusion. In other words, it shows that not all primitive/Seifert knots are twisted torus knot position. In fact, we give infinitely many primitive/Seifert knots that are not twisted torus knot position.

Keywords

knots;twisted torus knots;primitive curves;Seifert curves;proper power curves;primitive/Seifert knots

References

  1. J. Berge, A classification of pairs of disjoint nonparallel primitives in the boundary of a genus two handlebody, arXiv:0910.3038
  2. J. Berge, Private communication, (2012).
  3. J. Berge and S. Kang, The hyperbolic P/P, P/$SF_{d}$, and P/$SF_{m}$ knots in $S^{3}$, preprint.
  4. M. Cohen, W. Metzler, and A. Zimmerman, What Does a Basis of F(a,b) Look Like?, Math. Ann. 257 (1981), 435-445. https://doi.org/10.1007/BF01465865
  5. J. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebraic and Geometric Topology 3 (2003), 435-472. https://doi.org/10.2140/agt.2003.3.435
  6. R. P. Osborne and R. S. Stevens, Group Presentations Corresponding to Spines of 3-Manifolds II, Trans. Amer. Math. Soc. 234 (1977), 213-243.
  7. H. Zieschang, On Heegaard Diagrams of 3-Manifolds, Asterisque 163-164 (1988), 247-280.

Acknowledgement

Supported by : Chonnam National University