Non-integer surgery and branched double covers of alternating knots

Construction of Non-alternating Knots

We investigate the behaviour of Rasmussen’s invariant s under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into non-alternating knots.

Non-triviality of Generalized Alternating Knots

In this article, we consider alternating knots on a closed surface in the 3-sphere, and show that these are not parallel to any closed surface disjoint from the prescribed one.

Non-peripheral Ideal Decompositions of Alternating Knots

An ideal triangulation T of a hyperbolic 3-manifold M with one cusp is non-peripheral if no edge of T is homotopic to a curve in the boundary torus of M . For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of M . A planar projection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two of which are id...

On the Heegaard Floer Homology of Branched Double-covers

Let L ⊂ S be a link. We study the Heegaard Floer homology of the branched double-cover Σ(L) of S, branched along L. When L is an alternating link, ĤF of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E term is a suitable variant of Khovanov’s homology for the link L, convergi...

Alternating Knots