The theory of infinite multiplication has been studied in the case of the Hawaiian earring group, and has been seen to simplify the description of that group. In this paper we try to extend the theory of infinite multiplication to other groups and give a few examples of how this can be done. In particular, we discuss the theory as applied to symmetric groups and braid groups. We also give an equivalent definition to K. Eda's infinitary product as the fundamental group of a modified wedge product.

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)

A template is a key tool that we use to study knotted periodic orbits in the three-dimensional flow. The simplest type of template is the Lorenz template. In [5], Birman and Williams studied knotted periodic orbits with the aid of the Lorenz template. They discovered remarkable properties of Lorenz knots and links. No half twists exist in the Lorenz template. The new template is referred to be a Lorenz-like template when we add half twists. We looked at the template L(1,-1) in this paper, which has a positive half twist on the left-side and a negative half twist on the right. We look for the different types of knots and links that the template contains. Afterward, it was discovered that some knot types in L(1,-1) are fibered. Additionally, we look into the linking number of links in L(1,-1), as well as L(m; n) for m > 0 and n < 0. We have also explored the subtemplate of L(1,-1).

The anterior cruciate ligament (ACL) contributes to normal knee function, but it is commonly injured and has poor healing capabilities. Of the current treatments available for ACL reconstruction, none replicate the long-term mechanical properties of the ACL. It was hypothesized that tissue-engineered scaffolds comprised of reconstituted type I collagen fibers would have the potential to yield a more suitable treatment for ACL reconstruction. Ultra-violet (UV) radiation and 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide (EDC) were investigated as possible crosslinking methods for the scaffolds, and EDC crosslinking was deemed more appropriate given the gains in strength and stiffness afforded to individual collagen fibers. Scaffolds were composed of 54 collagen fibers, which were made using an extrusion process, organized in accordance with a braid-twist design; the addition of a hydrogel (gelatin) to this scaffold was also investigated. The scaffolds were tested mechanically to determine ultimate tensile strength (UTS), Young's modulus, and viscoelastic properties. Scaffolds were also evaluated for the cellular activity of primary rat lateral collateral ligament (LCL) and medial collateral ligament (MCL) fibroblast cells after 7, 14, and 21 days. The crosslinked scaffolds without gelatin exhibited mechanical and viscoelastic properties that were more similar to the human ACL. Cellular activity on the crosslinked scaffolds without gelatin was observed after 7 and 21 days, but no significant increase was observed with time. Although more studies are needed, these results indicate that a braid- twist scaffold (composed of collagen fibers) has the potential to serve as a scaffold for ACL replacement. / Master of Science

Seja $B_mM$ o grupo de tranças com $m$ cordas sobre uma superfície $M$ e seja $N$ uma subsuperfície de $M$. Estudaremos inicialmente condições necessárias e suficientes para as quais $B_nN$ é um subgrupo de $B_mM$ ($m$ podendo ser diferente de $n$), isto é, se considerarmos a inclusão $i\\colon N \\to M$, queremos estabelecer condições sobre $M$ e $N$ para que a aplicação induzida $i_\\ast \\colon B_nN \\to B_mM$ seja injetora. Em seguida, sob certas hipóteses para $N$ e $M$ calcularemos o comensurador, normalizador e centralizador de $B_nN$ em $B_mM$, sendo esse o objetivo principal desta dissertação. / Let $B_m(M)$ be the braid group with $m$ strings on a surface $M$ and let $N$ be a subsurface of $M$. We will study the necessary and sufficient conditions out of which $B_n(N)$ is a subgroup of $B_m(M)$ ($m$ can be different of $n$), it means, if we consider the inclusion $i \\colon N \\to M$, we would like to establish conditions for $M$ and $N$ for the induced application $i_\\ast \\colon B_nN \\to B_mM$ should be injective. After that, under some certain conditions for $M$ and $N$ we will calculate the commensurator, normalizer and centralizer of $Bn(N)$ in $Bm(M)$, being this one the principal objective of this work.

Seja $B_mM$ o grupo de tranças com $m$ cordas sobre uma superfície $M$ e seja $N$ uma subsuperfície de $M$. Estudaremos inicialmente condições necessárias e suficientes para as quais $B_nN$ é um subgrupo de $B_mM$ ($m$ podendo ser diferente de $n$), isto é, se considerarmos a inclusão $i\\colon N \\to M$, queremos estabelecer condições sobre $M$ e $N$ para que a aplicação induzida $i_\\ast \\colon B_nN \\to B_mM$ seja injetora. Em seguida, sob certas hipóteses para $N$ e $M$ calcularemos o comensurador, normalizador e centralizador de $B_nN$ em $B_mM$, sendo esse o objetivo principal desta dissertação. / Let $B_m(M)$ be the braid group with $m$ strings on a surface $M$ and let $N$ be a subsurface of $M$. We will study the necessary and sufficient conditions out of which $B_n(N)$ is a subgroup of $B_m(M)$ ($m$ can be different of $n$), it means, if we consider the inclusion $i \\colon N \\to M$, we would like to establish conditions for $M$ and $N$ for the induced application $i_\\ast \\colon B_nN \\to B_mM$ should be injective. After that, under some certain conditions for $M$ and $N$ we will calculate the commensurator, normalizer and centralizer of $Bn(N)$ in $Bm(M)$, being this one the principal objective of this work.

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Knot nullification is an unknotting operation performed on knots and links that can be used to model DNA recombination moves of circular DNA molecules in the laboratory. Thus nullification is a biologically relevant operation that should be studied. Nullification moves can be naturally grouped into two classes: coherent nullification, which preserves the orientation of the knot, and incoherent nullification, which changes the orientation of the knot. We define the coherent (incoherent) nullification number of a knot or link as the minimal number of coherent (incoherent) nullification moves needed to unknot any knot or link. This thesis concentrates on the study of such nullification numbers. In more detail, coherent nullification moves have already been studied at quite some length. This is because the preservation of the previous orientation of the knot, or link, makes the coherent operation easier to study. In particular, a complete solution of coherent nullification numbers has been obtained for the torus knot family, (the solution of the torus link family is still an open question). In this thesis, we concentrate on incoherent nullification numbers, and place an emphasis on calculating the incoherent nullification number for the torus knot and link family. Unfortunately, we were unable to compute the exact incoherent nullification numbers for most torus knots. Instead, our main results are upper and lower bounds on the incoherent nullification number of torus knots and links. In addition we conjecture what the actual incoherent nullification number of a torus knot will be.

In this thesis we study the homological behaviour of configuration spaces as the number of objects in the configuration goes to infinity. For unordered configurations of distinct points (possibly equipped with some internal parameters) in a connected, open manifold it is a well-known result, going back to G. Segal and D. McDuff in the 1970s, that these spaces enjoy the property of homological stability. In Chapter 2 we prove that this property also holds for so-called oriented configuration spaces, in which the points of a configuration are equipped with an ordering up to even permutations. There are two important differences from the unordered setting: the rate (or slope) of stabilisation is strictly slower, and the stabilisation maps are not in general split-injective on homology. This can be seen by some explicit calculations of Guest-Kozlowski-Yamaguchi in the case of surfaces. In Chapter 3 we refine their calculations to show that, for an odd prime p, the difference between the mod-p homology of the oriented and the unordered configuration spaces on a surface is zero in a stable range whose slope converges to 1 as p goes to infinity. In Chapter 4 we prove that unordered configuration spaces satisfy homological stability with respect to finite-degree twisted coefficient systems, generalising the corresponding result of S. Betley for the symmetric groups. We deduce this from a general “twisted stability from untwisted stability” principle, which also applies to the configuration spaces studied in the next chapter. In Chapter 5 we study configuration spaces of submanifolds of a background manifold M. Roughly, these are spaces of pairwise unlinked, mutually isotopic copies of a fixed closed, connected manifold P in M. We prove that if the dimension of P is at most (dim(M)−3)/2 then these configuration spaces satisfy homological stability w.r.t. the number of copies of P in the configuration. If P is a sphere this upper bound on its dimension can be increased to dim(M)−3.