Nonsimple Knots Research Articles

An Atlas of Legendrian Knots

We present an atlas of Legendrian knots in standard contact three-space. This gives a conjectural Legendrian classification for all knots with arc index at most 9, including alternating knots through seven crossings and nonalternating knots through nine crossings. Our method involves a computer search of grid diagrams and applies to transverse knots as well. The atlas incorporates a number of new, small examples of phenomena such as transverse nonsimplicity and nonmaximal nondestabilizable Legendrian knots, and gives rise to new infinite families of transversely nonsimple knots.

A family of transversely nonsimple knots

We apply knot Floer homology to exhibit an infinite family of transversely nonsimple prime knots starting with $10_{132}$. We also discuss the combinatorial relationship between grid diagrams, braids, and Legendrian and transverse knots in standard contact $\mathbb{R}^3$.

Transversely nonsimple knots

By proving a connected sum formula for the Legendrian invariant $\lambda_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots.

Tunnel Number One Genus One Non-Simple Knots

Tunnel Number One Genus One Non-Simple Knots

Companionship of knots and the Smith conjecture

This paper studies the Smith Conjecture in terms of H. Schubert’s theory of companionship of knots. Suppose J is a counterexample to the Smith Conjecture, i.e. is the fixed point set of an action of Z p {{\textbf {Z}}_p} on S 3 {S^3} . Theorem. Every essential torus in an invariant knot space C ( J ) C(J) of J is either invariant or disjoint from its translates. Since the companions of J correspond to the essential tori in C ( J ) C(J) , this often allows one to split the action among the companions and satellites of J. In particular: Theorem. If J is composite, then each prime factor of J is a counterexample, and conversely. Theorem. The Smith Conjecture is true for all cabled knots. Theorem. The Smith Conjecture is true for all doubled knots. Theorem. The Smith Conjecture is true for all cable braids. Theorem. The Smith Conjecture is true for all nonsimple knots with bridge number less than five. In addition we show: Theorem. If the Smith Conjecture is true for all simple fibered knots, then it is true for all fibered knots. Theorem. The Smith Conjecture is true for all nonfibered knots having a unique isotopy type of incompressible spanning surface.

Simple Knots Without Unique Minimal Surfaces

We construct infinitely many simple knots and infinitely many nonsimple knots, all of genus one, and all having both knotted and unknotted minimal spanning surfaces. The complements of the simple knots all contain closed incompressible surfaces of genus two.

Simple knots without unique minimal surfaces

We construct infinitely many simple knots and infinitely many nonsimple knots, all of genus one, and all having both knotted and unknotted minimal spanning surfaces. The complements of the simple knots all contain closed incompressible surfaces of genus two.