This paper begins with a basic overview of the key concepts of classical and virtual knot theory. After introductions to concepts such as knot diagrams, Reidemeister moves, and virtual links, the paper discusses the bikei algebraic structure and the fundamental bikei. The paper describes an algorithm that converts fundamental bikei presentations to matrix representations, and then completes the resulting matrices. These completed matrices can return the value of two link invariants.

Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots}, and flat virtual knots is the quotient of virtual knots which equates the real positive and negative crossings, and in this sense is complementary to classical knot theory within virtual knot theory. We classify flat virtual tangles with no closed components and give bases for its ``infinitesimal'' algebras. The classification of the former can be used as an invariant on virtual tangles with no closed components and virtual braids. In a subsequent paper, we will show that the infinitesimal algebras are the target spaces of any universal finite-type invariants on the respective variants of the flat virtual tangles.

Virtual knot theory, introduced by Kauffman, is a generalization of classical knot theory of interest because its finite-type invariant theory is potentially a topological interpretation of Etingof and Kazhdan's theory of quantization of Lie bi-algebras. Classical knots inject into virtual knots}, and flat virtual knots is the quotient of virtual knots which equates the real positive and negative crossings, and in this sense is complementary to classical knot theory within virtual knot theory. We classify flat virtual tangles with no closed components and give bases for its ``infinitesimal'' algebras. The classification of the former can be used as an invariant on virtual tangles with no closed components and virtual braids. In a subsequent paper, we will show that the infinitesimal algebras are the target spaces of any universal finite-type invariants on the respective variants of the flat virtual tangles.

Virtual knot theory is an extension of classical knot theory based on a combinatorial presentation of crossing information. The appropriate extensions of braid groups and string link monoids have also been studied. While some previously known knot invariants can be evaluated for virtual objects, entirely new techniques can also be used, for example, the concept of index of a crossing, and its resulting (Gaussian) parity theory. In general, a parity is a rule which assigns 0 or 1 to each crossing in a knot or link diagram. Recently, they have also been defined for virtual braids. Here, novel parities for knots, braids, and string links are defined, some of their applications are explored, most notably, defining a new subgroup of the virtual braid groups. / Thesis / Master of Science (MSc)

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)

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 will take a look at a branch of topology called knot theory. We will first look at what started the study of this field, classical knot theory. Knot invariants such as the Bracket polynomial and the Jones polynomial will be introduced and studied. We will then explore racks and biracks along with the axioms obtained from the Reidemeister moves. We will then move on to generalize classical knot theory to what is now known as virtual knot theory which was first introduced by Louis Kauffman. Finally, we take a look at a newer aspect of knot theory, twisted virtual knot theory and we defined new link invariants for twisted virtual biracks.