This thesis arose from a desire to better understand the structures of automorphism groups and inner automorphism groups of quandles. We compute and give the structure of the automorphism groups of all dihedral quandles. In their paper Matrices and Finite Quandles, Ho and Nelson found all quandles (up to isomorphism) of orders 3, 4, and 5 and determined their automorphism groups. Here we find the automorphism groups of all quandles of orders 6 and 7. There are, up to isomoprhism, 73 quandles of order 6 and 289 quandles of order 7.

The translocation of a biopolymer through a narrow pore exists in universal cellular processes, such as the translocations of nascent proteins through ribosome and the degradation of protein by ATP-dependent proteases. However, the molecular details of these translocation processes remain unclear. Using computer simulations we study the translocations of a ubiquitin-like protein into a pore. It shows that the mechanism of co-translocational unfolding of proteins through pores depends on the pore diameter, the magnitude of pulling force and on whether the force is applied at the N- or the C-terminus. Translocation dynamics depends on whether or not polymer reversal is likely to occur during translocation. Although it is of interest to compare the timescale of polymer translocation and reversal, there are currently no theories available to estimate the timescale of polymer reversal inside a pore. With computer simulations and approximate theories, we show how the polymer reversal depends on the pore size, r, and the chain length, N. We find that one-dimensional transition state theory (TST) using the polymer extension along the pore axis as a reaction coordinate adequately predicts the exponentially strong dependence of the reversal rate on r and N. Additionally, we find that the transmission factor (the ratio of the exact rate and the TST rate) has a much weaker power law dependence on r and N. Finite-size effects are observed even for chains with several hundred monomers. If metastable states are separated by high energy-barriers, transitions between them will be rare events. Instead of calculating the relative energy by studying those transitions, we can calculate absolute free energy separately to compare their relative stability. We proposed a method for calculating absolute free energy from Monte Carlo or molecular dynamics data. Additionally, the diffusion of a knot in a tensioned polymer is studied using simulations and it can be modeled as a one-dimensional free diffusion problem. The diffusion coefficient is determined by the number of monomers involved in a knot and its tension dependence shows a maximum due to two dominating factors: the friction from solvents and “local friction” from interactions among monomers in a compact knot. / text

Berge introduced knots that are primitive/primitive with respect to the standard genus 2 Heegaard surface, F, for the 3-sphere; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. The examples Dean worked with are among the twisted torus knots. In Chapter 3, we show that a given knot can have distinct primitive/Seifert representatives with the same surface slope. In Chapter 4, we show that a knot can also have a primitive/primitive and a primitive/Seifert representative that share the same surface slope. In Section 5.2, we show that these two results are part of the same phenomenon, the proof of which arises from the proof that a specific class of twisted torus knots are fibered, demonstrated in Section 5.1. / text

For DNA molecules, topological complexity occurs exclusively as the result of knotting or linking of the polynucleotide backbone. By contrast, while a few knots and links have been found within the polypeptide backbones of some protein structures, non-planarity can also result from the connectivity between a polypeptide chain and attached metal structures. In this thesis, we survey the known types of knots, links, and non-planar graphs in protein structures with and without including such bonds between proteins and metals. Then we present new examples of protein structures containing M\"obius ladders and other non-planar graphs as a result of these bound metal atoms. Finally, we propose hypothetical structures illustrating specific disulfide connectivities that would result in the key ring link, the Whitehead link and the 5_1 knot, the latter two of which have thus far not been identified within protein structures.

Knot theory is the mathematical study of knots. In this thesis we study knots and one of its applications in DNA. Knot theory sits in the mathematical field of topology and naturally this is where the work begins. Topological concepts such as topological spaces, homeomorphisms, and homology are considered. Thereafter knot theory, and in particular, knot theoretical invariants are examined, aiming to provide insights into why it is difficult to answer the question "How can we tell knots appart?". In knot theory invariants such as the bracket polynomial, the Jones polynomial and tricolorability are considered as well as other helpful results like Seifert surfaces. Lastly knot theory is applied to DNA, where it will shed light on how certain enzymes interact with the genome.

A major question in Knot Theory concerns the process of trying to determine when two knots are different. A knot invariant is a quantity (number, polynomial, group, etc.) that does not change by continuous deformation of the knot. One of the simplest invariant of knots is colorability. In this thesis, we study Fox colorings of knots and knots that are colored by linear Alexander quandles. In recent years, there has been an interest in reducing Fox colorings to a minimum number of colors. We prove that any Fox coloring of a 13-colorable knot has a diagram that uses exactly five colors. The ideas behind the reduction of colors in a Fox coloring is extended to knots colored by linear Alexander quandles. Thus, we prove that any knot colored by either the linear Alexander quandle Z5[t]/(t − 2) or Z5[t]/(t − 3) has a diagram using only four colors.

John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.

No description available.

In this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph. Furthermore, we devise a new quantum algorithm for approximating the phase of a unitary matrix. Finally, we provide quantum tools that can be utilized to extract the structure of black-box modules and algebras. While quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT) ) is highly constrained by the requirement of high-precision controlled phase shift operators, which remain difficult to realize. In the second part of this dissertation, we introduce an alternative approach to approximately implement QPE with arbitrary constantprecision controlled phase shift operators. The new quantum algorithm bridges the gap between QPE algorithms based on QFT and Kitaev’s original approach. For approximating the eigenphase precise to the nth bit, Kitaev’s original approach does not require any controlled phase shift operator. In contrast, QPE algorithms based on QFT or approximate QFT require controlled phase shift operators with precision of at least Pi/2n. The new approach fills the gap and requires only arbitrary constant-precision controlled phase shift operators. From a physical implementation viewpoint, the new algorithm outperforms Kitaev’s approach. iii The other problem we investigate relates to approximating the Tutte polynomial. We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e 2πi/5 ,e−2πi/5 ) is DQC1-complete and at points (q k , 1 + 1−q−k (q 1/2−q−1/2) 2 ) for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones-Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid ˜b such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of ˜b are alternating links. The final part of the dissertation focuses on finding the structure of a black-box module or algebra. Suppose we are given black-box access to a finite module M or algebra over a finite ring R, and a list of generators for M and R. We show how to find a linear basis and structure constants for M in quantum poly(log |M|) time. This generalizes a recent quantum algorithm of Arvind et al. which finds a basis representation for rings. We then show that iv our algorithm is a useful primitive allowing quantum computers to determine the structure of a finite associative algebra as a direct sum of simple algebras. Moreover, it solves a wide variety of problems regarding finite modules and rings. Although our quantum algorithm is based on Abelian Fourier transforms, it solves problems regarding the multiplicative structure of modules and algebras, which need not be commutative. Examples include finding the intersection and quotient of two modules, finding the additive and multiplicative identities in a module, computing the order of an module, solving linear equations over modules, deciding whether an ideal is maximal, finding annihilators, and testing the injectivity and surjectivity of ring homomorphisms. These problems appear to be exponentially hard classically.

<p>This thesis is an introduction to virtual knots and the forbidden moves, and the closely related classes of welded and unwelded knots. Extensions of the Jones polynomial and the knot group to the various knot types are considered. We also examine the operation of connected sum for virtual and welded knots, and we review the proof that every virtual knot can be untied using the forbidden moves.</p> / Master of Science (MSc)