Periodic projections of alternating knots

Alternating Knots

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...

Lissajous Knots and Knots with Lissajous Projections

Knots in R which may be parameterized by a single cosine function in each coordinate are called Lissajous knots. We show that twist knots are Lissajous knots if and only if their Arf invariants are zero. We further prove that all 2-bridge knots and all (3, q)-torus knots have Lissajous projections.

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.

Cusp Volumes of Alternating Knots

We show that the cusp volume of a hyperbolic alternating knot can be bounded above and below in terms of the twist number of an alternating diagram of the knot. This leads to diagrammatic estimates on lengths of slopes, and has some applications to Dehn surgery. Another consequence is that there is a universal lower bound on the cusp density of hyperbolic alternating knots.