There are several ways to describe juggling patterns mathematically using combinatorics and algebra. In my thesis I use these ideas to build a new system using braid groups. A new kind of graph arises that helps describe all braids that can be juggled.

This thesis consists of three applications of Ranicki's algebraic theory of surgery to the topology of manifolds. The common theme is a decomposition of a global algebraic object into simple local pieces which models the decomposition of a global topological object into simple local pieces. Part I: Algebraic reconstruction of 4-manifolds. We extend the product and glueing constructions for symmetric Poincaré complexes, pairs and triads to a thickening construction for a symmetric Poincaré representation of a quiver. Gay and Kirby showed that, subject to certain conditions, the fold curves and fibres of a Morse 2-function F : M4 → Ʃ 2 determine a quiver of manifold and glueing data which allows one to reconstruct M and F up to diffeomorphism. The Gay-Kirby method of reconstructing M glues the pre-images of disc neighbourhoods of cusps and crossings with thickenings of regular fibres and thickenings of cobordisms between regular fibres. We use our thickening construction for a symmetric Poincaré representation of a quiver to give an algebraic analogue of the Gay-Kirby result to reconstruct the symmetric Poincaré complex (C(M); ϕ M) of M from a Morse 2-function. Part II: The L-theory of triangular matrix rings. We construct a chain duality on the category of left modules over a triangular matrix ring A = (A1;A2;B) where A1;A2 are rings with involution and B is an (A1;A2)-bimodule. We describe the resulting L-theory of A and relate it to the L-theory of A1;A2 and to the change of rings morphism B ⊗A2 − : A2-Mod → A1-Mod. By examining algebraic surgery over A we define a relative algebraic surgery operation on an (n+1)-dimensional symmetric Poincaré pair with data an (n+2)-dimensional triad. This gives an algebraic model for a half-surgery on a manifold with boundary. We then give an algebraic analogue of Borodzik, Némethi and Ranciki's half-handle decomposition of a relative manifold cobordism and show that every relative Poincaré cobordism is homotopy equivalent to a union of traces of elementary relative surgeries. Part III: Seifert matrices of braids with applications to isotopy and signatures. Let β be a braid with closure ^β a link. Collins developed an algorithm to find the Seifert matrix of the canonical Seifert surface Ʃ of ^ β constructed by Seifert's algorithm. Motivated by Collins' algorithm and a construction of Ghys, we define a 1-dimensional simplicial complex K(β) and a bilinear form λβ : C1(K(β);Z)×C1(K(β);Z) → Z[ 1/2 ] such that there is an inclusion K(β) ~ → Ʃ which is a homotopy equivalence inducing an isomorphism H1(Ʃ;Z) ≅ H1(K(β);Z) such that [λβ] : H1(K(β);Z) × H1(K(β);Z) → Z ⊂ Z[ 1/2 ] is the Seifert form of Ʃ. We show that this chain level model is additive under the concatenation of braids and then verify that this model is chain equivalent to Banchoff's combinatorial model for the linking number of two space polygons and Ranicki's surgery theoretic model for a chain level Seifert pairing. We then define the chain level Seifert pair (λβ; d β) of a braid β and equivalence relations, called A and Â-equivalence. Two n-strand braids are isotopic if and only if their chain level Seifert pairs are A-equivalent and this yields a universal representation of the braid group. Two n-strand braids have isotopic link closures in the solid torus D2 ×S1 if and only if their chain level Seifert pairs are  A-equivalent and this yields a representation of the braid group modulo conjugacy. We use the first representation to express the ω signature of a braid β in terms of the chain level Seifert pair (λ β; d β).

No description available.

A large number of people throughout the world have naturally kinky hair that may be very difficult to manage. These people often subject their hair to vigorous and harsh treatment processes in order to straighten it and hence make it more manageable. Hair braiding is a popular and fashionable trend amongst many people, in particular people of African descent. Braided hairstyles serve to preserve hair and protect it, and to give it time to rejuvenate after a period of harsh treatment. During the braiding process synthetic hair is attached to natural hair by weaving a length of the natural hair into one end of each braid. Other materials like wool or cotton may be use used to achieve different hairstyles and textures. Several strands of natural hair are used to secure each braid. The braids are normally left intact for a number of weeks or even months. Although braiding is a helpful African hair grooming practice, the process of taking down or detangling the braids is labor intensive and entails each braid being cut just below where the natural hair ceases and the natural hair being untangled from the braid using a safety pin, a needle or a fine toothed comb. The labor and long hours required to detangle braided hairstyles often results in braid wearers frustratingly pulling on their braided hair. This behavior inevitably destroys the hair follicle and leaves the hair damaged. According to a study conducted by the University of Cape Town’s dermatology department, braiding may be the root cause of traction alopecia (TA) amongst braid wearers. Traction alopecia is a form of alopecia, or gradual hair loss that is caused primarily by excessive pulling forces applied to the hair. The purpose of this current study was to investigate the factors, other than braid tightness, that affect the way and ease with which braids are detangled from the human hair. The study hypothesized that frictional forces present in braided hair were amongst these factors. It was hypothesized that introducing a lubricating formulation in the braids would allow for easier braid detangling. In order to decrease the prevalence of traction alopecia from braided hair, two hair strengthening actives were included in the test formulation. The study investigated the effects of the test formulations on braid detangling, hair friction and on the tensile strength of human hair. The study found that the method used did not pick up any significant differences between the braid detangling forces of treated braids when compared to the braid vi detangling forces of untreated hair. The same method used to measure braid detangling forces was able to show that there are variations in the braid detangling forces of different sections along the braid length. The method to measure braid detangling was based on the principles of hair combability measurements. The study also found that although the method used to measure braid detangling forces was unsuccessful in picking up significant differences in braid detangling forces of treated hair and untreated hair, the method used to measure the frictional forces of human hair showed that the frictional forces of hair treated with test formulations were significantly different than that of untreated hair. The method used to measure frictional forces was based on the capstan approach. The Capstan method measures the forces required to slide a weighted hair fibre over a curved surface of reference material. The interaction between the weighted fibre and the reference material simulates the movement of hair out of a braid ensemble in the braid detangling process. The optimum mixture with the minimum coefficient of friction, predicted a coefficient of friction of 0.61 ± 0.04. The optimum formulation was found to be one that contained 30% Cyclopentasiloxane , 0% PEG-12 Dimethicone, 10% 18-MEA, 29% water, 10% hair strengthening actives, 12.86% emulsifier combination and 8% other oils. The study also showed that including hair strengthening actives, such as hydrolysed proteins had significant effects in the tensile strength properties of chemically treated African hair.

Thesis advisor: Julia E. Grigsby / We use the Birman-Ko-Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element δ are fibered. We classify links which admit such a braid representative in geometric terms as boundaries of plumbings of positive Hopf bands to a disk. Rudolph constructed fibered strongly quasipositive links as closures of positive words on certain generating sets of Bₙ and we prove that Rudolph’s condition is equivalent to ours. We compute the sutured Khovanov homology groups of positive braid closures in homological degrees i = 0,1 as sl₂(ℂ)-modules. Given a condition on the sutured Khovanov homology of strongly quasipositive braids, we show that the sutured Khovanov homology of the closure of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element agrees with that of positive braid closures in homological degrees i ≤ 1 and show this holds for the class of such braids on three strands. / Thesis (PhD) — Boston College, 2017. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

This thesis studies matter emergent as topological excitations of quantum geometry in quantum gravity models. In these models, states are framed four-valent spin networks embedded in a topological three manifold, and the local evolution moves are dual Pachner moves. We first formulate our theory of embedded framed four-valent spin networks by proposing a new graphic calculus of these networks. With this graphic calculus, we study the equivalence classes and the evolution of these networks, and find what we call 3-strand braids, as topological excitations of embedded four-valent spin networks. Each 3-strand braid consists of two nodes that share three edges that may or may not be braided and twisted. The twists happen to be in units of 1/3. Under certain stability condition, some 3-strand braids are stable. Stable braids have rich dynamics encoded in our theory by dual Pachner moves. Firstly, all stable braids can propagate as induced by the expansion and contraction of other regions of their host spin network under evolution. Some braids can also propagate actively, in the sense that they can exchange places with substructures adjacent to them in the graph under the local evolution moves. Secondly, two adjacent braids may have a direct interaction: they merge under the evolution moves to form a new braid if one of them falls into a class called actively interacting braids. The reverse of a direct interaction may happen too, through which a braid decays to another braid by emitting an actively interacting braid. Thirdly, two neighboring braids may exchange a virtual actively interacting braid and become two different braids, in what is called an exchange interaction. Braid dynamics implies an analogue between actively interacting braids and bosons. We also invent a novel algebraic formalism for stable braids. With this new tool, we derive conservation laws from interactions of the braid excitations of spin networks. We show that actively interacting braids form a noncommutative algebra under direction interaction. Each actively interacting braid also behaves like a morphism on non-actively interacting braids. These findings reinforce the analogue between actively interacting braids and bosons. Another important discovery is that stable braids admit seven, and only seven, discrete transformations that uniquely correspond to analogues of C, P, T, and their products. Along with this finding, a braid's electric charge appears to be a function of a conserved quantity, effective twist, of the braids, and thus is quantized in units of 1/3. In addition, each $CPT$-multiplet of actively interacting braids has a unique, characteristic non-negative integer. Braid interactions turn out to be invariant under C, P, and T. Finally, we present an effective description, based on Feynman diagrams, of braid dynamics. This language manifests the analogue between actively interacting braids and bosons, as the topological conservation laws permit them to be singly created and destroyed and as exchanges of these excitations give rise to interactions between braids that are charged under the topological conservation rules. Additionally, we find a constraint on probability amplitudes of braid interactions. We discuss some subtleties, open issues, future directions, and work in progress at the end.

This thesis studies matter emergent as topological excitations of quantum geometry in quantum gravity models. In these models, states are framed four-valent spin networks embedded in a topological three manifold, and the local evolution moves are dual Pachner moves. We first formulate our theory of embedded framed four-valent spin networks by proposing a new graphic calculus of these networks. With this graphic calculus, we study the equivalence classes and the evolution of these networks, and find what we call 3-strand braids, as topological excitations of embedded four-valent spin networks. Each 3-strand braid consists of two nodes that share three edges that may or may not be braided and twisted. The twists happen to be in units of 1/3. Under certain stability condition, some 3-strand braids are stable. Stable braids have rich dynamics encoded in our theory by dual Pachner moves. Firstly, all stable braids can propagate as induced by the expansion and contraction of other regions of their host spin network under evolution. Some braids can also propagate actively, in the sense that they can exchange places with substructures adjacent to them in the graph under the local evolution moves. Secondly, two adjacent braids may have a direct interaction: they merge under the evolution moves to form a new braid if one of them falls into a class called actively interacting braids. The reverse of a direct interaction may happen too, through which a braid decays to another braid by emitting an actively interacting braid. Thirdly, two neighboring braids may exchange a virtual actively interacting braid and become two different braids, in what is called an exchange interaction. Braid dynamics implies an analogue between actively interacting braids and bosons. We also invent a novel algebraic formalism for stable braids. With this new tool, we derive conservation laws from interactions of the braid excitations of spin networks. We show that actively interacting braids form a noncommutative algebra under direction interaction. Each actively interacting braid also behaves like a morphism on non-actively interacting braids. These findings reinforce the analogue between actively interacting braids and bosons. Another important discovery is that stable braids admit seven, and only seven, discrete transformations that uniquely correspond to analogues of C, P, T, and their products. Along with this finding, a braid's electric charge appears to be a function of a conserved quantity, effective twist, of the braids, and thus is quantized in units of 1/3. In addition, each $CPT$-multiplet of actively interacting braids has a unique, characteristic non-negative integer. Braid interactions turn out to be invariant under C, P, and T. Finally, we present an effective description, based on Feynman diagrams, of braid dynamics. This language manifests the analogue between actively interacting braids and bosons, as the topological conservation laws permit them to be singly created and destroyed and as exchanges of these excitations give rise to interactions between braids that are charged under the topological conservation rules. Additionally, we find a constraint on probability amplitudes of braid interactions. We discuss some subtleties, open issues, future directions, and work in progress at the end.

Neste trabalho apresentamos uma descrição geométrica do grupo de tranças no disco Bpnq e sua apresentação em termos de geradores e relatores no famoso teorema da apresentação de Artin. Mostraremos também que o grupo de tranças puras PBpnq, grupo que possui a permutação trivial das cordas, é bi-ordenável, ou seja, exibiremos uma ordenação para PBpnq que será invariante pela multiplicação em ambos os lados. Esse processo é dado a partir da combinação da técnica de pentear Artin e a expansão Magnus para grupos livres. / In this work, we present a geometric description of the braids groups of the disk Bpnq and its presentation in terms of generators and relations in the famous theorem of Artin\'s presentation. We also show that groups of pure braids, denoted by PBpnq, groups that have the trivial permutation of the strings, are bi-orderable, that is, we will present the explicit construction of a strict total ordering of pure braids PBpnq which is invariant under multiplying on both sides. This process is given from the combination of the techniques of combing Artin and Magnus expansion to free groups.

Motivated by reductionism's current inability to encompass the quantum theory we explore an indivisible and dynamical wholeness as an underlying foundation for physics. After reviewing the role of wholeness in the quantum theory we set a philosophical background aiming at introducing an ontology, based on a dynamical wholeness. Equipped with the philosophical background we then propose a mathematical realization by representing the dynamics with a non-trivial elementary embedding from the mathematical universe to itself. By letting the embedding interact with itself through application we obtain a left-distributive universal algebra that is isomorphic to special braids. Via the connection between braids and quantum and statistical physics we show that a the mathematical structure obtained from wholeness yields known physics in a special case. In particular we point out the connections to algebras of observables, spin networks, and statistical mechanical models used in solid state physics, such as the Potts model. Furthermore we discuss the general case and there the possibility of interpreting the mathematical structure as a dynamics beyond unitary evolution, where entropy increase is involved.

Neste trabalho apresentamos uma descrição geométrica do grupo de tranças no disco Bpnq e sua apresentação em termos de geradores e relatores no famoso teorema da apresentação de Artin. Mostraremos também que o grupo de tranças puras PBpnq, grupo que possui a permutação trivial das cordas, é bi-ordenável, ou seja, exibiremos uma ordenação para PBpnq que será invariante pela multiplicação em ambos os lados. Esse processo é dado a partir da combinação da técnica de pentear Artin e a expansão Magnus para grupos livres. / In this work, we present a geometric description of the braids groups of the disk Bpnq and its presentation in terms of generators and relations in the famous theorem of Artin\'s presentation. We also show that groups of pure braids, denoted by PBpnq, groups that have the trivial permutation of the strings, are bi-orderable, that is, we will present the explicit construction of a strict total ordering of pure braids PBpnq which is invariant under multiplying on both sides. This process is given from the combination of the techniques of combing Artin and Magnus expansion to free groups.