PRIME KNOTS WITH ARC INDEX UP TO 11 AND AN UPPER BOUND OF ARC INDEX FOR NON-ALTERNATING KNOTS

ARC INDEX OF PRETZEL KNOTS OF TYPE (−p, q, r)

Let D be a diagram of a knot or a link L. Suppose that there is a simple closed curve C meeting D in k distinct points which divide D into k arcs α1, α2, . . . , αk with the following properties: (1) Each αi has no self-crossing. (2) If αi crosses over αj at a crossing, then i > j and it crosses over αj at any other crossings with αj . (3) For each i, there exists an embedded disk di such that ...

Enumerating the Prime Alternating Knots, Part Ii

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.

Alternating Knots

Non-triviality of Generalized Alternating Knots

In this article, we consider alternating knots on a closed surface in the 3-sphere, and show that these are not parallel to any closed surface disjoint from the prescribed one.