The tunnel numbers of all 11- and 12-crossing alternating knots

Crossing Number of Alternating Knots in S × I

A hundred years ago, Tait conjectured that the number of crossings in a reduced alternating projection of an alternating knot is minimal. This statement was proven in 1986 by Kauffman, Murasugi and Thistlethwaite, [6], [10], [11], working independently. Their proofs relied on the new polynomials generated in the wake of the discovery of the Jones polynomial. We usually think of this result as a...

Classification of Alternating Knots with Tunnel Number One

An alternating diagram encodes a lot of information about a knot. For example, if an alternating knot is composite, this is evident from the diagram [10]. Also, its genus ([3], [12]) and its crossing number ([7], [13], [17]) can be read off directly. In this paper, we apply this principle to alternating knots with tunnel number one. Recall that a knot K has tunnel number one if it has an unknot...

Alternating Knots

All Exceptional Surgeries on Alternating Knots Are Integral Surgeries

We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.

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...