Semiadequacy Properties for the Edge Jones Polynomial Coefficients

It had been known, essentially by definition of semiadequacy, that the leading/trailing terms of the Jones polynomial of a semiadequate link must be monic. But this relationship was further deepened with formulas for the second and third coefficient. These formulas were found independently by Dasbach-Lin. We explain these formulas, and then survey some applications, including:

1) the construction of odd crossing number amphicheiral knots,

2) the construction of infinitely many positive braid links which admit no minimal string positive braid representative,

3) a construction of minimal crossing almost alternating diagrams for almost all crossing numbers,

4) the construction of infinitely many pseduo-alternating links which are not homogeneous.