Upper and lower bound on delta-crossing number and tabulation of knots

TL;DR

This paper improves bounds on the delta-crossing number of knots, relates it to the triple-crossing number, and provides tables of minimal delta-diagrams for prime knots up to four crossings, advancing knot classification methods.

Contribution

It strengthens bounds on the delta-crossing number, links it to the triple-crossing number, and generates comprehensive tables for prime knots up to four crossings.

Findings

Established new bounds on delta-crossing numbers.

Obtained triple-crossing numbers for specific knots.

Generated tables of minimal delta-diagrams for prime knots.

Abstract

We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a few knots. We also prove that we can always find at least one tangle from the set of four tangles, in any triple-crossing projections of any non-trivial knot or non-split link. In the last section, we enumerate and generate tables of minimal delta-diagrams for all prime knots up to the delta-crossing number equal to four. We also give a concise survey about known inequalities between integer-valued classical knot invariants.