Abstract:
The knotting probability of an arc diagram is defined as the quadruplet of four kinds of finner knotting probabilities which are invariant under a reasonable deformation containing an isomorphism on an arc diagram. Then it is shown that every oriented spatial arc admits three kinds of arc diagrams unique up to isomorphisms determined from the spatial arc and the projection, so that the knotting probability of a spatial arc is defined. The definition of the knotting probability of and arc diagram uses the fact that every arc diagram induces a unique chord diagram representing a ribbon 2-knot. Then the knotting probability of an arc diagram is set to measure how many non-trivial ribbon genus 2 surface-knots occur from the chord diagram induced from the arc diagram. The condition for an arc diagram with the knotting probability 0 and the condition for an arc diagram with the knotting probability 1 are given together with some other properties and some examples.
References:
1) A. Kawauchi, Knotting probability of an arc diagram
2) A. Kawauchi, Unique diagram of a spatial arc and the knotting probability
