UC Davis

This dissertation is based on the following three publications of the author [15,16,17], focusing on quandles and their applications in knot theory.Chapter 2 defines a family of quandles and studies their algebraic invariants. The axioms of a quandle imply that the columns of its Cayley table are permutations. The chapter studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and Hom quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number. Chapter 3 describes the triple point number of non-orientable surface-links using symmetric quandles. Analogous to a classical knot diagram, a surface-link can be generically projected to 3-space and given crossing information to create a broken sheet diagram. The triple point number of a surface-link is the minimal number of triple points among all broken sheet diagrams that lift to that surface-link. The chapter generalizes a family of Oshiro to show that there are non- split surface-links of arbitrarily many trivial components whose triple point number can be made arbitrarily large. Chapter 4 focuses on the triple point number of surface-links found in Yoshikawa’s table. Yoshikawa made an enumeration of knotted surfaces in R4 with ch-index 10 or less. This remark- able table is the first to tabulate knotted surfaces analogous to the classical prime knot table. This chapter compiles the known triple point numbers of the surface-links represented in Yoshikawa’s table and calculates or provides bounds on the triple point number of the remaining surface-links. Chapter 5 is included to study quandle invariants of knotoids. The chapter focuses on the chirality of knotoids using shadow quandle colorings and the shadow quandle cocycle invariant. The shadow coloring number and the shadow quandle cocycle invariant is shown to distinguish infinitely many knotoids from their mirrors. Specifically, the knot-type knotoid 31 is shown to be chiral. The weight of a quandle 3-cocycle is used to calculate the crossing numbers of infinitely many multi-linkoids.