This paper contains some further applications of the study of knot diagrams by genus. Introducing a procedure of regularization for knot generators, and using invariants derived from the Jones polynomial (degrees, congruences, and the Fiedler–Polyak–Viro Gauß diagram formulas for its Vassiliev invariants), we examine the existence of genus-minimizing diagrams for almost alternating and almost positive knots. In particular, we examine the existence of such knots such that either all or none of their almost alternating/positive diagrams have the minimal genus property. We prove that the genus of almost positive non-split links is determined by the Alexander polynomial.
"Minimal genus of links and fibering of canonical surfaces." Illinois J. Math. 59 (2) 399 - 448, Summer 2015. https://doi.org/10.1215/ijm/1462450708