On Knots with Nontrivial Interpolating Manifolds

Nontrivial Alexander Polynomials of Knots and Links

In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguishing knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial knots. Furthermore we prove that these invariants decide if a genus one knot ...

Dehn surgeries on knots in product manifolds

We show that if a surgery on a knot in a product sutured manifold yields the same product sutured manifold, then this knot is a 0or 1-crossing knot. The proof uses techniques from sutured manifold theory.

Dehn Surgery on Knots in 3-manifolds

It has been known for over 30 years that every closed connected orientable 3manifold is obtained by surgery on a link in S [8]. However, a classification of such 3-manifolds in terms of this surgery construction has remained elusive. This is due primarily to the lack of uniqueness of the surgery description. In [5], Kirby gave us a calculus of surgery diagrams. However, the lack of a ‘canonical...

Framed Knots in 3-manifolds

For a fixed isotopy type K of unframed knots in S there are infinitely many isotopy classes of framed knots that correspond to K when we forget the framing. We show that the same fact is true for all the isotopy types of unframed knots in a closed oriented 3-manifold M , provided that M 6= (S × S)#M . On the other hand for any M = (S × S)#M ′ we construct examples of isotopy classes of unframed...

Four - manifolds , geometries and knots