Integer inequalities between knot invariants, skein tree depth and delta-crossing numbers

The maximum length of the shortest path from a leaf to the root of a skein tree for knots and links gives a measure of the complexity of computing link polynomials by the skein relation (the Jones polynomial, the Alexander-Conway polynomial, and more generally the HOMFLY-PT polynomial). We combine theoretical and computational results on the skein tree depth of knots and links. We prove a new upper bound on the skein tree depth of a link and give examples of links where the new bound is stronger than the known bound; we also give a lower bound. Moreover, we derive tables of knots and links with their skein tree depth that were up to now undetermined (for some of them, we provide a range of possible values). The paper surveys known (and new) inequalities between integer-valued classical knot invariants. It features a visual graph of the relations.