# Monodromy Groups on Knot Surgery 4-manifolds

Yun, Ki-Heon

• Accepted : 2013.12.04
• Published : 2013.12.23
• 19 8

#### Abstract

In the article we show that nondieomorphic symplectic 4-manifolds which admit marked Lefschetz fibrations can share the same monodromy group. Explicitly we prove that, for each integer g > 0, every knot surgery 4-manifold in a family {$E(2)_K{\mid}K$ is a bered 2-bridge knot of genus g in $S^3$} admits a marked Lefschetz fibration structure which has the same monodromy group.

#### Keywords

knot surgery 4-manifold;Lefschetz fibration;monodromy group

#### References

1. R. Fintushel and R. Stern, Nondiffeomorphic symplectic 4-manifolds with the same Seiberg-Witten invariants, Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol., Coventry, 1999, pp. 103-111(electronic).
2. David Gabai and William H. Kazez, Pseudo-Anosov maps and surgery on fibred 2-bridge knots, Topology Appl. 37(1)(1990), 93-100. https://doi.org/10.1016/0166-8641(90)90018-W
3. R. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142(3)(1995), 527-595. https://doi.org/10.2307/2118554
4. Y. Gurtas, Positive Dehn Twist Expressions for some New Involutions in Mapping Class Group, 2004, arXiv:math.GT/0404310.
5. A. Kas, On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math. 89(1)(1980), 89-104. https://doi.org/10.2140/pjm.1980.89.89
6. A. Kawauchi, A survey of knot theory, Birkhauser Verlag, Basel, 1996, Translated and revised from the 1990 Japanese original by the author.
7. Y. Matsumoto, Lefschetz fibrations of genus two-a topological approach, Topology and Teichmuller spaces (Katinkulta, 1995), World Sci. Publishing, River Edge, NJ, 1996, pp. 123-148.
8. J. Park and K.-H. Yun, Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds, Math. Ann. 345(3)(2009), 581-597. https://doi.org/10.1007/s00208-009-0366-0
9. J. Park and K.-H. Yun, Lefschetz fibration structures on knot surgery 4-manifolds, Michigan Math. J. 60(3)(2011), 525-544. https://doi.org/10.1307/mmj/1320763047
10. J. Park and K.-H. Yun, Families of nondi eomorphic 4-manifolds with the same seiberg-witten invariants, arXiv1302.0587 [math.GT] (2013). To appear in J. Symplectic Geom.
11. K.-H. Yun, On the signature of a Lefschetz fibration coming from an involution, Topology Appl. 153(12)(2006), 1994-2012. https://doi.org/10.1016/j.topol.2005.07.007
12. G. Burde and H. Zieschang, Knots, second ed., de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 2003.
13. R. Fintushel and R. Stern, Knots, links, and 4-manifolds, Invent. Math. 134(1998), no. 2, 363-400. https://doi.org/10.1007/s002220050268
14. K.-H. Yun, Twisted fiber sums of Fintushel-Stern's knot surgery 4-manifolds, Trans. Amer. Math. Soc. 360(11)(2008), 5853-5868. https://doi.org/10.1090/S0002-9947-08-04623-0
15. R. Gompf and A. Stipsicz, 4-manifolds and Kirby calculus, American Mathematical Society, Providence, RI, 1999.
16. M. Korkmaz, Noncomplex smooth 4-manifolds with Lefschetz fibrations, IMRN(3) (2001), 115-128.
17. R. Fintushel and R. Stern, Families of simply connected 4-manifolds with the same Seiberg-Witten invariants, Topology 43(6)(2004), (6), 1449-1467. https://doi.org/10.1016/j.top.2004.03.002

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)