J. Math. Sci. Univ. Tokyo 12 (2005), 105–110. Gauge Theory and the A-Polynomial

Abstract. In this article, we explain how to use instanton Floer homology of various Dehn surgeries along knots in integer homology spheres to prove that their A-polynomial is non-trivial. In particular, we show that all non-trivial knots in S3 have non-trivial A-polynomial. Not long ago, Kronheimer and Mrowka gave a gauge theoretic proof of Property P for knots in S3. In fact, the proof found in [11] establishes much more: all (1/n)–surgeries along a non-trivial knot in S3 admit a rep-resentation of their fundamental group in SU(2) with non-abelian image. Their work has been used in [2] by Boyer and Zhang to show that the A– polynomial of any non-trivial knot in S3 is non-trivial. In this short note we give a gauge-theoretic proof of this fact and various generalizations, the approach being closer in spirit to the results of Kronheimer and Mrowka contained in [11] and [12] since we use holonomy perturbations that natu-rally arise in gauge theory. Let us begin by recalling a crucial step in the proof of Property P: one must establish that the 0–surgery of S3 along K has non-vanishing Floer homology HF∗(Y0(K)). These Floer groups are generated by non-degenerate flat connections on the SO(3)–bundle P over Y0(K) with non-trivial second Steifel-Whitney class. If the moduli space of flat connections M(Y0(K)) is degenerate, holonomy perturbations of the flatness equation are used to define the Floer chain groups. The explicit construction of Floer homology for Y0(K) is not important for our purpose, only matters the fact that if the moduli space M(Y0(K)) is empty, then HF∗(Y0(K)) is trivial. The 0–surgery being obtained by Dehn filling the knot complement YK along the longitude λK, the moduli space M(Y0(K)) on P such that w2(P) = 0 can be obtained from the moduli space of irreducible flat SU(2)– connections over the knot complement M∗(YK) by taking flat connection