Ki-Heon Yun

Twisted fiber sums of Fintushel-Stern’s knot surgery 4-manifolds

In the article, we study Fintushel-Sterns knot surgery four-manifold E(n) K and its monodromy factorization. For fibered knots we provide a smooth classification of knot surgery 4-manifolds up to twisted fiber sums. We then show that other constructions of 4-manifolds with the same Seiberg-Witten invariants are in fact diffeomorphic.

Monodromy Groups on Knot Surgery 4-manifolds

Abstract. In the article we show that nondi eomorphic symplectic 4-manifolds whichadmit marked Lefschetz brations can share the same monodromy group. Explicitlywe prove that, for each integer g > 0, every knot surgery 4-manifold in a familyfE(2) K jK is a bered 2-bridge knot of genus g in S 3 g admits a marked Lefschetz bra-tion structure which has the same monodromy group. 1. IntroductionSeiberg-Witten invariants are one of the most powerful invariants in the classi- cation of smooth 4-manifolds and Fintushel-Stern’s knot surgery method is one ofthe most e ective methods to modify smooth structures on a given 4-manifold. ButSeiberg-Witten invariants are not complete invariants and there are known examplesof nondi eomorphic symplectic 4-manifolds which share the same Seiberg-Witteninvariants [3, 15].R. Fintushel and R. Stern showed that Seiberg-Witten invariants of knot surgery4-manifoldE(2) K = E(2)] F=m K S 1 (M K S 1 )can be computed by using the Alexander polynomial of the related knot K[2]. If werestrict our attention to a bered knot K, then E(2)

MULTI-VARIABLE ALEXANDER POLYNOMIAL OF A PLAT

In the article we study the multi-variable Alexander polynomial of a link in a plat form or in a closed braid form. By using the method, we find an algorithm how to compute the multi-variable Alexander polynomial of the 2a-fold dihedral cover and the a-fold irregular cover of a two bridge knot K(a,b).

RATIONAL BLOW-DOWNS AND NONSYMPLECTIC 4-MANIFOLDS WITH ONE BASIC CLASS

We present a simple way to construct an infinite family of simply connected, nonspin, smooth 4-manifolds with one basic class which do not admit a symplectic structure with either orientation.