This paper gives a brief introduction into the fundaments of knot theory: introducing knot diagrams, knot invariants, and two techniques to determine whether or not two knots are ambient isotopic. After discussing the basics of knot theory an algebraic coloring of knots knows as a bikei is introduced. The algebraic structure as well as the various axioms that define a bikei are defined. Furthermore, an extension between the Alexander polynomial of a knot and the Alexander Bikei is made. The remainder of the paper is devoted to reintroducing a modified homology and cohomology theory for involutory biquandles known as bikei, first introduced in [18]. The bikei 2-cocycles can be utilized to enhance the counting invariant for unoriented knots and links as well as unoriented and non-orienteable knotted surfaces in R4.

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This thesis introduces a new enhancement for virtual birack counting invariants. We first introduce knots and other general types of knots (oriented knots, framed knots, racks, and biracks). Then we’ll discuss the methods, knot invariants, mathematicians use to identify whether two knots are different. Next we’ll look at knots with virtual crossings and knots with a good involution. Finally, we introduce a new symmetric enhancement for virtual birack counting invariants and provide an example.

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.

We begin by introducing knots and links generally and identifying various geometric, polynomial, and integer-based knot and link invariants. Of particular importance to this paper are ternary operations and Niebrzydowski tribrackets defined in [12], [10]. We then introduce multi-tribrackets, ternary algebraic structures following the specified region coloring rules with di↵erent operations at multi-component and single component crossings. We will explore examples of each of the invariants and conclude with remarks on the direction of the introduced multi-tribracket theory.

A hybrid molecular dynamics/multiparticle collision dynamics algorithm was constructed to model the pressure-driven flow of a compressible fluid through a nanoscopic channel of square cross-sectional area, as well as the effect of this flow on the configuration of a polymer chain that was tethered to the surface of this nanochannel. In the process of simulating channel flow, a new adiabatic partial slip boundary condition was created as well as a modified source/sink inlet and outlet boundary condition that could maintain a specified pressure gradient across the channel without the large entrance effects typically associated with these algorithms. The results of the flow simulations were contrasted with the results from a series solution to the Navier-Stokes equation for isothermal compressible flow, and showed excellent agreement with the results from the series solution when slip-boundary conditions were applied. A finitely extendible non-linear elastic spring and bead polymer chain was used to simulate the effect of flow on the polymer chain configuration under poor solvent and θ solvent conditions. Under θ solvent conditions, the cyclical dynamics that have been previousy observed for tethered polymer chains in pure shear flows were noted, however they were restricted to the end of the polymer chain. Under poor solvent conditions, the polymer adopted a metastable helix configuration as it collapsed to a globule state. The study also examined interchain and intrachain entanglements in polymers using the granny knot and overhand knot. The mechanisms by which these tangles untied themselves were determined. At low flow rates, the tangles unravelled by the end of the chain migrating through the loops of the tangle. At high flow rates, the tangles behaved like an entrained object as they reptated towards the end of the chain.

The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links.

A hybrid molecular dynamics/multiparticle collision dynamics algorithm was constructed to model the pressure-driven flow of a compressible fluid through a nanoscopic channel of square cross-sectional area, as well as the effect of this flow on the configuration of a polymer chain that was tethered to the surface of this nanochannel. In the process of simulating channel flow, a new adiabatic partial slip boundary condition was created as well as a modified source/sink inlet and outlet boundary condition that could maintain a specified pressure gradient across the channel without the large entrance effects typically associated with these algorithms. The results of the flow simulations were contrasted with the results from a series solution to the Navier-Stokes equation for isothermal compressible flow, and showed excellent agreement with the results from the series solution when slip-boundary conditions were applied. A finitely extendible non-linear elastic spring and bead polymer chain was used to simulate the effect of flow on the polymer chain configuration under poor solvent and θ solvent conditions. Under θ solvent conditions, the cyclical dynamics that have been previousy observed for tethered polymer chains in pure shear flows were noted, however they were restricted to the end of the polymer chain. Under poor solvent conditions, the polymer adopted a metastable helix configuration as it collapsed to a globule state. The study also examined interchain and intrachain entanglements in polymers using the granny knot and overhand knot. The mechanisms by which these tangles untied themselves were determined. At low flow rates, the tangles unravelled by the end of the chain migrating through the loops of the tangle. At high flow rates, the tangles behaved like an entrained object as they reptated towards the end of the chain.

In this thesis, we study Legendrian and transverse isotopy problem for cabled knot types. We give two structural theorems to describe when the (r,s)- cable of a Legendrian simple knot type K is also Legendrian simple. We then study the same problem for cables of the positive trefoil knot. We give a complete classification of Legendrian and transverse cables of the positive trefoil. Our results exhibit many new phenomena in the structural understanding of Legendrian and transverse knots. we then extend these results to the other positive torus knots. The key ingredient in these results is to find necessary and sufficient conditions on maximally thickened contact neighborhoods of the positive torus knots in three sphere.

In this dissertation we prove that if an n-stranded pretzel knot K has an essential Conway sphere, then there exists an Alexander grading s such that the rank of knot Floer homology in this grading, [mathematical equation], is at least two. As a consequence, we are able to easily classify pretzel knots admitting L-space surgeries. We conjecture that this phenomenon occurs more generally for any knot in S³ with an essential Conway sphere. We also exhibit an infinite family of knots, each of which admits a nontrivial genus two mutant which shares the same total dimension of knot Floer homology, while being distinguished by knot Floer homology as a bigraded invariant. Additionally, the genus two mutation interchanges the [mathematical symbol]-graded knot Floer homology groups in [mathematical symbol]-gradings k and -k. This infinite family of examples supports a second conjecture, namely that the total rank of knot Floer homology is invariant under genus two mutation. / text