Kauffman–Harary conjecture for alternating virtual knots

A Conjecture of Kauuman on Amphicheiral Alternating Knots

We give a counterexample to the following conjecture of Kauuman 2]: Conjecture Let K be an amphicheiral alternating knot. Then there exists a reduced alternating knot-diagram D of K, such that G(D) is isomorphic to G (D), where G(D) is a checkerboard-graph of D and G (D) its dual.

Alternating Knots

Simon’s conjecture for fibered knots

We prove Simon’s conjecture for fibered knots in S: a fibered knot group surjects at most finitely many other knot groups.

Alternating Quadrisecants of Knots

It is known [Pann, Kup] that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and the knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. I show that every (nontrivial, tame) knot in R has an alternating quadrisecant. T...

Rogers-ramanujan Type Identities for Alternating Knots

We highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogers-Ramanujan type identities for alternating knots as conjectured by Garoufalidis, Lê and Zagier.