[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.5831/HMJ.2013.35.4.775

PRIMITIVE/SEIFERT KNOTS WHICH ARE NOT TWISTED TORUS KNOT POSITION

Kang, Sungmo (Department of Mathematics Education, Chonnam National University)

Publication Information

Honam Mathematical Journal / v.35, no.4, 2013 , pp. 775-791 More about this Journal

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;

Citations & Related Records

Reference

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3	H. Zieschang, On Heegaard Diagrams of 3-Manifolds, Asterisque 163-164 (1988), 247-280.
4	J. Berge, A classification of pairs of disjoint nonparallel primitives in the boundary of a genus two handlebody, arXiv:0910.3038
5	J. Berge, Private communication, (2012).
6	J. Berge and S. Kang, The hyperbolic P/P, P/ $SF_{d}$ , and P/ $SF_{m}$ knots in $S^{3}$ , preprint.
7	M. Cohen, W. Metzler, and A. Zimmerman, What Does a Basis of F(a,b) Look Like?, Math. Ann. 257 (1981), 435-445. DOI