## An obstruction to sliceness via contact geometry and “classical” gauge theory.(English)Zbl 0843.57011

The author shows that if a knot has a non-negative maximal Thurston-Bennequin invariant, then it is not slice. The method is a combination of 3-dimensional contact geometry and 4-dimensional “classical” gauge theory à la Donaldson. This is supposed to be in contrast to the new gauge theoretic techniques of Kronheimer and Mrowka which lead to the proof of the Thom conjecture and the Bennequin inequality for the slice genus of a knot. This inequality implies the first mentioned result. As a nice corollary of his result the author mentions that the simplest Casson-handle (with one kink in every step) is non-standard if all kinks have the same sign. This result was first obtained by Ž. Bižaca and is conjectured for arbitrary signs of the kinks.
Reviewer: P.Teichner (Mainz)

### MSC:

 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)

### Keywords:

knots; slice knots; gauge theory; contact structures
Full Text:

### References:

 [1] D. Bennequin: Entrelacements et équations de Pfaff. Astérisque108-8 (1983) 87-161 · Zbl 0573.58022 [2] ?. Bi?aca: A family of exotic Casson handles, to appear Proc. A.M.S. [3] M. Boileau, L. Rudolph, Plombage et structures de contact (in preparation) [4] S. Donaldson: Complex curves and surgery. Inst. Hautes Études Sci. Publ. Math.68 (1988) (1989) 91-97. · Zbl 0696.57007 [5] Y. Eliashberg: Legendrian and transversal knots in tight contact 3-manifolds (preprint 1991) · Zbl 0809.53033 [6] Y. Eliashberg: Classification of overtwisted contact structures on 3-manifolds. Invent. Math.98 (1989) 623-637 · Zbl 0684.57012 [7] T. Erlandsson: Geometry of contact transformations in dimension three. Ph. D. Thesis, Uppsala, 1981 [8] P. Kronheimer, T. Mrowka: Gauge theory for embedded surfaces. I. Topology32 (1993) 773-826 · Zbl 0799.57007 [9] N. Kuhn: A conjectural inequality on the slice genus of links. Ph. D. Thesis, Princeton University, 1984 [10] H. Lyon: Torus knots in the complements of links and surfaces. Mich. Math. J.27 (1980) 39-46 · Zbl 0423.57004 [11] W. Menasco: A proof of the Bennequin-Milnor unknotting conjecture (preprint 1993) [12] J. Milnor: Morse Theory, Annals of Mathematics Studies, Number 51, Princeton University Press, Princeton, N. J. 1969 [13] L. Rudolph: Braided surfaces and Seifert ribbons for closed braids. Comm. Math. Hel.58 (1983) 1-37 · Zbl 0522.57017 [14] L. Rudolph: Constructions of quasipositive knots and links, II, Four-Manifold Theory (Contemp. Math. 35; C. Gordon and R. Kirby, eds.), AMS, 1984, pp. 485-491 · Zbl 0604.57003 [15] L. Rudolph: A congruence between link polynomials, Math. Proc. Camb. Phil. Soc.107 (1990) · Zbl 0703.57005 [16] L. Rudolph:A characterization of quasipositive Seifert surfaces (Constructions of quasipositive knots and links, III). Topology31 (1992) 231-237 · Zbl 0763.57008 [17] L. Rudolph: Quasipositive annuli (Constructions of quasipositive knots and links, IV), J. Knot Theory Ramif.1 (1992) 451-466. · Zbl 0773.57006 [18] L. Rudolph: Totally tangential links of intersection of complex plane curves with round spheres, Topology’90 (B. Apanasov et al., eds.), de Gruyter, 1992, pp. 343-349 · Zbl 0785.57002 [19] L. Rudolph:Quasipositivity as an obstruction to sliceness. Bull. Am. Math. Soc.29 (1993) 51-59 · Zbl 0789.57004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.