On positivities of links: an investigation of braid simplification and defect of Bennequin inequalities

A knot can be envisioned as a piece of hyper-elastic string in space, interwoven about itself, with its ends tied together. One of the most fundamental questions asked by the knot theorist is: when are two knots different? That is, when will no amount of bending and sliding suffice to transform one knot into another? The standard technique of the knot theorist in answering this question is to define properties, called knot invariants, which serve to (partially) characterize the knot: if two knots differ in some invariant, they cannot be the same.

In this thesis we investigate a family of invariants, called positivities. To understand various forms of positivity, it must first be understood that a knot, presented as an object in space, is a very unwieldy thing. As such, knots are often presented via some complexity-reducing technique, such as a projection onto a 2-dimensional plane. In projecting onto a plane, depth is lost, and crossings—points where one strand of the knot passes over another strand—appear. When the knot is oriented (i.e., endowed with a coherent sense of “forward” and “backward”, each crossing takes on one of two signs: either the understrand points left of the overstrand, in which case the crossing is positive, or it points to the right, in which case the crossing is negative. When all crossings of a diagram are positive, we say that the knot is positive. Depending on the rigidity we require of the 2-dimensional knot representation, we can define various forms of positivity. This thesis makes heavy use of a particular form of 2-dimensional knot representation, called a braid, in order to study positivities. In particular, we study so-called quasipositivity and strong quasipositivity, which are positivities defined using braid representatives of knots. We establish several new criteria, as well as two new knot invariants, which can detect the presence of (strong) quasipositivity.