No abstract

We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsváth, Stipsicz, and Szabó. For the upper bound, we use a known bound for braid index 3 and a new bound for braid index 4. Both bounds coincide, so that we obtain a sharp result.

Abstract. There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.

We call a knot in the 3-sphere SU (2)-simple if all representations of the fundamental group of its complement which map a meridian to a tracefree element in SU (2) are binary dihedral. This is a generalization of being a 2-bridge knot. Pretzel knots with bridge number ≥ 3 are not SU (2)-simple. We provide an infinite family of knots K with bridge number ≥ 3 which are SU (2)-simple.One expects the instanton knot Floer homology I (K) of a SU (2)-simple knot to be as small as it can be -of rank equal to the knot determinant det(K). In fact, the complex underlying I (K) is of rank equal to det(K), provided a genericity assumption holds that is reasonable to expect. Thus formally there is a resemblance to strong L-spaces in Heegaard Floer homology. For the class of SU (2)-simple knots that we introduce this formal resemblance is reflected topologically: The branched double covers of these knots are strong L-spaces. In fact, somewhat surprisingly, these knots are alternating. However, the Conway spheres are hidden in any alternating diagram.With the methods we use, we obtain the result that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group. This makes use of a non-vanishing result of Kronheimer-Mrowka. 2 RAPHAEL ZENTNER homology to the Alexander polynomial, established independently by Kronheimer and Mrowka in [20] and Lim in [24].Proposition 7.3. If a knot K is SU (2)-simple and satisfies the genericity hypothesis 7.2, then its instanton Floer chain complex CI π (K) has no non-zero differentials and is of total rank det(K). In particular, the total rank of reduced instanton knot Floer homology I (K) is also equal to det(K).Denoting Y (T (p, q)) the complement of a tubular neighborhood of the torus knot T (p, q), we may glue Y (T (p, q)) and Y (T (r, s)) together along their boundary torus in such a way that a meridian of the first torus knot is mapped to a Seifert fibre of the second and vice versa. It is a result of Motegi [30] that these 3-manifolds Y = Y (T (p, q), T (r, s)) are SO(3)-cyclic, i.e. admit only cyclic SO(3) representations of their fundamental group. Using the concept of strongly invertible knots, we obtain Theorem 4.14. The 3-manifold Y (T (p, q), T (r, s)) comes with an involution with quotient S 3 . It is a branched double cover of some knot or 2-component link L(T (p, q), T (r, s)) in S 3 , well defined up to mutation by the involution on either side of the essential torus in Y (T (p, q), T (r, s)). If pqrs − 1 is odd then L(T (p, q), T (r, s) is a knot. If in addition both T (p, q) and T (r, s) are non-trivial torus knots, then the knot L(T (p, q), T (r, s) is SU (2)-simple, but is not a 2-bridge knot.We give an explicit description of the knots L(T (p, q), T (r, s)) as a decomposition of two tangles in Section 6 below. In fact, each tangle is explicitly described in Theorem 6.1. Somewhat to our surprise, we have obtained Theorem 6.5. The knots L(T (p, q), T (r, s)) are alternating. The Conway sphere giving rise to the essen...

We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N ) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic of the moduli space is equal to the quantum sl(N ) polynomial of the graph evaluated at unity. Possible extensions of the result are also indicated.