RASMUSSEN INVARIANTS OF SOME 4-STRAND PRETZEL KNOTS

Abstract

Abstract. It is known that there is an in nite family of generalpretzel knots, each of which has Rasmussen s-invariant equal to thenegative value of its signature invariant. For an instance, homo-logically ˙-thin knots have this property. In contrast, we nd anin nite family of 4-strand pretzel knots whose Rasmussen invariantsare not equal to the negative values of signature invariants. 1. IntroductionKhovanov [7] introduced a graded homology theory for oriented knotsand links, categorifying Jones polynomials. Lee [10] de ned a variant ofKhovanov homology and showed the existence of a spectral sequence ofrational Khovanov homology converging to her rational homology. Leealso proved that her rational homology of a knot is of dimension two.Rasmussen [13] used Lee homology to de ne a knot invariant sthat isinvariant under knot concordance and additive with respect to connectedsum. He showed that s(K) = ˙(K) if Kis an alternating knot, where˙(K) denotes the signature of K.Suzuki [14] computed Rasmussen invariants of most of 3-strand pret-zel knots. Manion [11] computed rational Khovanov homologies of allnon quasi-alternating 3-strand pretzel knots and links and found theRasmussen invariants of all 3-strand pretzel knots and links.For general pretzel knots and links, Jabuka [5] found formulas fortheir signatures. Since Khovanov homologically ˙-thin knots have sequal to ˙, Jabuka’s result gives formulas for sinvariant of any quasi-alternating pretzel knot. Note that quasi-alternating pretzel knots are