UC Berkeley

In addition to classical knot theory, low-dimensional topology is also concerned with a variety of alternative classes of knot-like objects, including knotoids, braids, and string links. In this thesis we explore several of these theories as well as certain connections between them. While our results cover several different types of structures, they nonetheless build on each other.

The theory of knotoids is an extension of classical knot theory whereby knot diagrams are allowed to have two endpoints instead of being a closed loop. One of the most important knotoid invariants is the height h, which measures how far a knotoid is from being a classical knot. After defining the signed versions h_+ and h_- of the height, we prove that together they determine the unsigned height by the simple formula h_+ + h_- = h, and we demonstrate a few applications.

String links, which are tangles that have been normalized with respect to the locations of the endpoints of the strands, come with a natural monoid operation that generalizes both connected summation of knots and composition of braids. Previous work on strings links has shown that the string link monoids are cancellative up to multiplication with units. Here we strengthen that result by proving the freeness of unit multiplication on either side, thus answering a question of Blair–Burke–Koytcheff and implying full cancellativity of the string link monoids. In the same section, we prove a weaker version of the well-known cosmetic crossing conjecture and explore applications of knotoids to braid groups.

We also study an infinite version of string links, which we call Z-tangles. These Z-tangles are interesting in part because a version of Birman–Hilden theory applies to Z-tangles as they relate to knotoids. Specifically, there is a natural forgetful function from the set of knotoids to the set of Z-tangles up to isotopy. We give several pieces of evidence for the conjectural injectivity of this Birman–Hilden map, most notably by using some of the results of earlier chapters to identify classes of knotoids that will each be uniquely determined by their image.