<p>Slice-ribbon conjecture has been proved for some special families of knots. In this thesis, we briefly mention some of these results.</p> / Master of Science (MSc)

The problem under consideration is that of determining a simple and effective invariant of knots. To this end, the Conway polynomial is defined as a generalization of Alexander's original knot polynomial. It is noted, however, that the Conway polynomial is not a complete invariant. If two knots are equivalent, as defined in this investigation, then they receive identical polynomials. Yet, if two knots have identical polynomials, no information about their equivalence may be obtained. To define the Conway polynomial, the Axioms for Computation are given and many examples of their use are included. A major result of this investigation is the proof of topological invariance of these polynomials and the proof that the axioms are sufficient for the calculation of the knot polynomial for any given knot or link.

A knot is an embedding of S1 in three-dimensional space. Generally, it can be thought of as a knotted piece of string with the ends glued together. When we project a knot into the plane, we can create a knot diagram in which we specify which portion of the string lies on top at each place that the string crosses itself. To perform a crossing change on a knot, one can imagine cutting one portion of the string at a crossing, allowing another portion of the string to pass through, and then gluing the cleaved ends back together. We define the distance between two knots, K1 and K2, to be the minimum number of crossing changes one must perform on either K1 or K2 to obtain the other knot. Circular DNA can become knotted during biological processes such as recombination and replication. We can model knotted DNA with a mathematical knot. Type II topoisomerases are the enzymes tasked with keeping DNA unknotted, and they act on double-stranded circular DNA by breaking the backbone of the DNA, allowing another segment of DNA to pass through, and then re-sealing the break. Thus, performing a crossing change on a knot models the action of this protein. Specifically, studying knots of distance one can help us better understand how the action of a type II topisomerase on double-stranded circular DNA can alter DNA topology. We create a knot distance graph by letting the set of vertices be rational knots with up to and including thirteen crossings and by placing an edge between two vertices if the two knots corresponding to those vertices are of distance one. A neighborhood of a vertex, v, in a graph is the set of vertices with which v is adjacent via an edge. Using graph theoretical and topological tools, we examine graphs of knot distances and define a mapping between distance one neighborhoods. Additionally, this idea can also be examined and visualized as performing Dehn surgery on the double branched cover of a knot.

This thesis reviews the history of knot theory with an emphasis on the diagrammatic approach to studying knots. Also covered are the basic concepts and notions of graph theory and how these two fields are related with an example of a knot diagram and how to associate it to a graph.

In this thesis, we show that every periodic virtual knot can be realized as the closure of a periodic virtual braid. If K is a q-periodic virtual knot with quotient K_*, then the knot group G_{K_*} is a quotient of G_K and we derive an explicit q-symmetric Wirtinger presentation for G_K, whose quotient is a Wirtinger presentation for G_{K_*}. When K is an almost classical knot and q=p^r, a prime power, we show that K_* is also almost classical, and we establish a Murasugi-like congruence relating their Alexander polynomials modulo p. This result is applied to the problem of determining the possible periods of a virtual knot $K$. For example, if K is an almost classical knot with nontrivial Alexander polynomial, our result shows that K can be p-periodic for only finitely many primes p. Using parity and Manturov projection, we are able to apply the result and derive conditions that a general q-periodic virtual knot must satisfy. The thesis includes a table of almost classical knots up to 6 crossings, their Alexander polynomials, and all known and excluded periods. / Thesis / Doctor of Philosophy (PhD)

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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)

No description available.

No description available.

Doctor of Philosophy / Department of Mathematics / David Yetter / In this paper we give a new generalization of the Khovanov homology. The construction begins with a Frobenius-algebra-like object in a category of graded vector-spaces with an anyonic braiding, with most of the relations weaken to hold only up to phase. The construction of Khovanov can be adapted to give a new link homology theory from such data. Both Khovanov's original theory and the odd Khovanov homology of Oszvath, Rassmusen and Szabo arise from special cases of the construction in which the braiding is a symmetry.