This thesis is largely expository, and we provide a survey on unknotting operations. We examine these local transformations for classical, virtual and welded knots and use their properties to calculate upper bounds on unknotting numbers. In addition, the thesis contains some original work, such as the definition and properties of the algebraic unknotting numbers of virtual and welded knots, an algebraic reformulation of t4-conjecture, and a new method to tell if a knot can be turned into a torus knot with one crossing change. / Thesis / Master of Science (MSc)

This thesis provides a partial classification of all 3-stranded pretzel knots K=P(p,q,r) with unknotting number one. Scharlemann-Thompson, and independently Kobayashi, have completely classified those knots with unknotting number one when p, q, and r are all odd. In the case where p=2m, we use the signature obstruction to greatly limit the number of 3-stranded pretzel knots which may have unknotting number one. In Chapter 3 we use Greene's strengthening of Donaldson's Diagonalization theorem to determine precisely which pretzel knots of the form P(2m,k,-k-2) have unknotting number one, where m is an integer, m>0, and k>0, k odd. In Chapter 4 we use Donaldson's Diagonalization theorem as well as an unknotting obstruction due to Ozsv\'ath and Szab\'o to partially classify which pretzel knots P(2,k,-k) have unknotting number one, where k>0, odd. The Ozsv\'ath-Szab\'o obstruction is a consequence of Heegaard Floer homology. Finally in Chapter 5 we explain why the techniques used in this paper cannot be used on the remaining cases. / text

Knot nullification is an unknotting operation performed on knots and links that can be used to model DNA recombination moves of circular DNA molecules in the laboratory. Thus nullification is a biologically relevant operation that should be studied. Nullification moves can be naturally grouped into two classes: coherent nullification, which preserves the orientation of the knot, and incoherent nullification, which changes the orientation of the knot. We define the coherent (incoherent) nullification number of a knot or link as the minimal number of coherent (incoherent) nullification moves needed to unknot any knot or link. This thesis concentrates on the study of such nullification numbers. In more detail, coherent nullification moves have already been studied at quite some length. This is because the preservation of the previous orientation of the knot, or link, makes the coherent operation easier to study. In particular, a complete solution of coherent nullification numbers has been obtained for the torus knot family, (the solution of the torus link family is still an open question). In this thesis, we concentrate on incoherent nullification numbers, and place an emphasis on calculating the incoherent nullification number for the torus knot and link family. Unfortunately, we were unable to compute the exact incoherent nullification numbers for most torus knots. Instead, our main results are upper and lower bounds on the incoherent nullification number of torus knots and links. In addition we conjecture what the actual incoherent nullification number of a torus knot will be.

Adams conjectured that unknotting tunnels of tunnel number 1 manifolds are always isotopic to a geodesic. We generalize this question to tunnel number n manifolds. We find that there exist complete hyperbolic structures and a choice of spine of a compression body with genus 1 negative boundary and genus n ≥ 3 outer boundary for which (n−2) edges of the spine self-intersect. We use this to show that there exist finite volume one-cusped hyperbolic manifolds with a system of n tunnels for which (n−1) of the tunnels are homotopic to geodesics arbitrarily close to self-intersecting. This gives evidence that the generalization of Adam's conjecture to tunnel number n ≥ 2 manifolds may be false.