# TWISTED TORUS KNOTS WITH GRAPH MANIFOLD DEHN SURGERIES

• Kang, Sungmo
• Received : 2015.02.17
• Published : 2016.01.31
• 32 3

#### Abstract

In this paper, we classify all twisted torus knots which are doubly middle Seifert-fibered. Also we show that all of these knots possibly except a few admit Dehn surgery producing a non-Seifert-fibered graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. This provides another infinite family of knots in $S^3$ admitting Dehn surgery yielding such manifolds as done in [5].

#### Keywords

knots;Dehn surgery;graph manifolds;Seifert curves;twisted torus knots;R-R diagrams

#### 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 and S. Kang, Primitive and Seifert curves in a genus two handlebody, preprint.
3. J. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003), 435-472. https://doi.org/10.2140/agt.2003.3.435
4. W. Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology, 39-98, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968.
5. S. Kang, Knots in $S^3$ admitting graph manifold Dehn surgeries, J. Korean Math. Soc. 51 (2014), no. 6, 1221-1250. https://doi.org/10.4134/JKMS.2014.51.6.1221
6. S. Kang, Knots admitting Seifert-fibered surgeries over $S^2$ with four exceptional fibers, Bull. Korean Math. Soc. 52 (2015), no. 1, 313-321. https://doi.org/10.4134/BKMS.2015.52.1.313
7. R. P. Osborne and R. S. Stevens, Group presentations corresponding to spines of 3-manifolds II, Trans. Amer. Math. Soc. 234 (1977), no. 1, 213-243.