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Title: Jones-type link invariants and applications to 3-manifold topology
Creator: Lee, Christine Ruey Shan
Date: 2015
Collection: Electronic Theses & Dissertations
Description: "It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semi-adequate link has a well-defined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semi-adequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new...
Show more"It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semi-adequate link has a well-defined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semi-adequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new family of pretzel knots which are not adequate. We also study the relationship between the Jones polynomial and the topology of the knot complement by relating its coefficients to the non-orientable genus of an alternating knot. The two-sided bound we obtain can often determine the non-orientable genus. Lastly, we develop the connection between the Jones polynomial and a knot invariant coming from Heegaard Floer homology by using it to classify 3-braids which are L-space knots." -- Abstract.
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Title: A combinatorial approach to knot theory : volume bounds for hyperbolic semi-adequate link complements
Creator: Giambrone, Adam Joseph
Date: 2014
Collection: Electronic Theses & Dissertations
Description: An interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semi-adequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semi-adequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed...
Show moreAn interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semi-adequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semi-adequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed braids, and families of links that have both plat and closed braid aspects. By more closely studying each of these families, we can often improve the lower bounds on volume provided by the main result. Furthermore, we show that the bounds on volume can be expressed in terms of a single stable coefficient of the colored Jones polynomial. By doing this, we provide new collections of links that satisfy a Coarse Volume Conjecture. The main approach of this entire work is to use a combinatorial perspective to study the connections among knot theory, hyperbolic geometry, and graph theory.
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Title: Topics in knot theory : on generalized crossing changes and the additivity of the Turaev genus
Creator: Balm, Cheryl Lyn Jaeger
Date: 2013
Collection: Electronic Theses & Dissertations
Description: We first study cosmetic crossing changes and cosmetic generalized crossing changes in knots of genus one, satellite knots, and knots obtained via twisting operations on standardly embedded tori in the knot complement. As a result, we find obstructions to the existence of cosmetic generalized crossing changes in several large families of knots. We then study Turaev surfaces and use decomposing spheres to analyze the additivity of the Turaev genus for the summands of composite knots with Turaev...
Show moreWe first study cosmetic crossing changes and cosmetic generalized crossing changes in knots of genus one, satellite knots, and knots obtained via twisting operations on standardly embedded tori in the knot complement. As a result, we find obstructions to the existence of cosmetic generalized crossing changes in several large families of knots. We then study Turaev surfaces and use decomposing spheres to analyze the additivity of the Turaev genus for the summands of composite knots with Turaev genus one.
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Title: The Homology Polynomial and the Burau Representation for Pseudo-Anosov Braids
Creator: Shultz, Warren Michael
Date: 2021
Collection: Electronic Theses & Dissertations
Description: The homology polynomial is an invariant for pseudo-Anosov mapping classes \cite{Birman2010}. We study the homology polynomial as an invariant for pseudo-Anosov braids and its connection to the Burau representation. Given a pseudo-Anosov braid $\beta\in B_n$, we determine necessary and sufficient conditions under which the homology polynomial of $\beta$ is equal to the the characteristic polynomial of the image of $\beta$ under the Burau representation. In particular, we build upon \cite...
Show moreThe homology polynomial is an invariant for pseudo-Anosov mapping classes \cite{Birman2010}. We study the homology polynomial as an invariant for pseudo-Anosov braids and its connection to the Burau representation. Given a pseudo-Anosov braid $\beta\in B_n$, we determine necessary and sufficient conditions under which the homology polynomial of $\beta$ is equal to the the characteristic polynomial of the image of $\beta$ under the Burau representation. In particular, we build upon \cite{Band2007} and show that the orientation cover associated to a pseudo-Anosov braid is equivalent to a quotient to the Burau cover when the measured foliations associated to $\beta$ have odd-ordered singularities at each puncture and any singularity that occurs in the interior of $D_n$ is even-ordered. We next construct an algorithm which allows us to determine the homology polynomial from the Burau representation for an arbitrary pseudo-Anosov braid. As an application, we show how to easily determine the homology polynomial for large family of pseudo-Anosov braids.
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Title: Volumes, determinants, and meridian lengths of hyperbolic links
Creator: Burton, Stephan D., 1987-
Date: 2017
Collection: Electronic Theses & Dissertations
Description: We study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of non-alternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi...
Show moreWe study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of non-alternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi log det(K)/c(K). We prove that the closure of the set of determinant densities contains the set [0, v_8]. We examine the conjecture, due to Champanerkar, Kofman, and Purcell that vol(K) < 2 pi log det (K) for alternating hyperbolic links, where vol(K) = vol(S^3\ K) is the hyperbolic volume and det(K) is the determinant of K. We prove that the conjecture holds for 2-bridge links, alternating 3-braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of K exceeds some function of the twist number of K.We derive bounds on the length of the meridian and the cusp volumeof hyperbolic knots in terms of the topology of essential surfaces spanned by the knot.We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound.As applications we find knot diagrammatic upper bounds on the meridian length and the cusp volume of hyperbolic adequate knots and we obtain new large families of knots withmeridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.
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Title: Topics in link homology
Creator: Jaeger, Thomas Constantin
Date: 2011
Collection: Electronic Theses & Dissertations
Description: We prove two results about mutation invariance of link homology theories: Weshow that Khovanov's universal sl(2) homology is invariant under mutationand that the reduced sl(n) homology defined by Khovanov and Rozansky isinvariant under component-preserving positive mutation when n is odd. Wealso give a relationship between the Khovanov homology of a closed positive3-braid and the Khovanov homology of the braid after adding a number of fulltwists.

Title: Asymptotics of the Turaev-Viro invariants and their connections in low-dimensional topology
Creator: Kumar, Sanjay Lakshman
Date: 2021
Collection: Electronic Theses & Dissertations
Description: We study the Turaev-Viro invariants of 3-manifolds as well as their relationship to invariants arising from hyperbolic geometry. We first construct a closed formula for the Turaev-Viro invariants for hyperbolic once-punctured torus bundles. We then examine a conjecture by Chen and Yang which states that the asymptotics of the Turaev-Viro invariants recover the hyperbolic volume of the manifold. Using topological tools, we are able to construct infinite families of new examples of hyperbolic...
Show moreWe study the Turaev-Viro invariants of 3-manifolds as well as their relationship to invariants arising from hyperbolic geometry. We first construct a closed formula for the Turaev-Viro invariants for hyperbolic once-punctured torus bundles. We then examine a conjecture by Chen and Yang which states that the asymptotics of the Turaev-Viro invariants recover the hyperbolic volume of the manifold. Using topological tools, we are able to construct infinite families of new examples of hyperbolic links in the 3-sphere which satisfy the conjecture. Additionally, we show a general method for augmenting a link such that the resulting link has hyperbolic complement which satisfies the conjecture. As an application of the constructed links in the 3-sphere, we extend the class of known examples which satisfy a conjecture by Andersen, Masbaum, and Ueno relating the quantum representations of surface mapping class groups to its Nielsen-Thurston classification. Most notably, we provide explicit elements in the mapping class group for a genus zero surface with n boundary components, for n > 3, as well as elements in the mapping class group for any genus g surface with four boundary components such that the obtained elements satisfy the conjecture. This is accomplished by utilizing the intrinsic relationship between the Turaev-Viro invariants and the quantum representations shown by Detcherry and Kalfagianni.
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Title: Diagrammatic and Geometric Invariants of Hyperbolic Weakly Generalized Alternating Knots
Creator: Bavier, Brandon
Date: 2021
Collection: Electronic Theses & Dissertations
Description: We study the relationship between knot and link diagrams on surfaces and their invariants coming from hyperbolic geometry. A \indef{link diagram on a surface}, denoted $\pi(L)\subset F$, is a way of projecting a link $L$ in some manifold $M$ onto some surface $F\subset M$. This generalizes the notion of a link diagram, and has been studied with a variety of conditions. We will work with the weakly generalized alternating knots and links of Howie and Purcell, which gives certain criteria to...
Show moreWe study the relationship between knot and link diagrams on surfaces and their invariants coming from hyperbolic geometry. A \indef{link diagram on a surface}, denoted $\pi(L)\subset F$, is a way of projecting a link $L$ in some manifold $M$ onto some surface $F\subset M$. This generalizes the notion of a link diagram, and has been studied with a variety of conditions. We will work with the weakly generalized alternating knots and links of Howie and Purcell, which gives certain criteria to ensure the diagram is interesting. A cusp of a hyperbolic knot $K$ is a neighborhood of $K$ in $M\setminus K$, and the cusp volume is the Euclidean volume of a maximal cusp. We show that the cusp volume of a weakly generalized alternating knot, with some additional conditions, is bounded both above and below based on the twist number of $\pi(K)$ and $\chi(F)$. This is done by constructing a new essential surface for $K$ that has nice properties, including having Euler characteristic based on the twist number as opposed to the crossing number. Our bound on cusp volume leads to interesting bounds on other geometric properties of $K$, including slope length and volumes of Dehn surgery. The volume}of a hyperbolic link $L$ in a manifold $M$ is the hyperbolic volume of the complement $M\setminus L$. We can show that volume is also bounded below by the twist number of $\pi(L)$ and $\chi(F)$. We do this by generalizing the Jones polynomial to weakly generalized alternating links, and showing that there is a relation between this polynomial and the twist number, and this polynomial and the volume. Through the course of proving this bound, we also get relations between the guts of the checkerboard surfaces of $L$ and this generalized Jones polynomial. In addition, if we are working inside a thickened surface, the twist number becomes an isotopy invariant of $L$.
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Title: Knot theory of Morse-Bott critical loci
Creator: Ozsarfati, Metin
Date: 2018
Collection: Electronic Theses & Dissertations
Description: We give an alternative proof of that a critical knot of a Morse-Bott function f : S3 → R is a graph knot where the critical set of f is a link in S3. Our proof inducts on the number of index-1 critical knots of f. -- Abstract.

Title: Invariants of topological and Legendrian links in lens spaces with a universally tight contact structure
Creator: Cornwell, Christopher R.
Date: 2011
Collection: Electronic Theses & Dissertations
Description: In this thesis a HOMFLY polynomial is found for knots and links in a lens space L(p, q). Further study of this polynomial invariant finds a relationship with the classical invariants of Legendrian and transverse links, when L(p, q) is endowed with a universally tight contact structure. In fact certain criteria are found which, if satisfied by any numerical invariant of links in L(p, q), guarantee that the invariant fits into a Bennequin type inequality. A linear function of the degree of the...
Show moreIn this thesis a HOMFLY polynomial is found for knots and links in a lens space L(p, q). Further study of this polynomial invariant finds a relationship with the classical invariants of Legendrian and transverse links, when L(p, q) is endowed with a universally tight contact structure. In fact certain criteria are found which, if satisfied by any numerical invariant of links in L(p, q), guarantee that the invariant fits into a Bennequin type inequality. A linear function of the degree of the HOMFLY polynomial is then shown to satisfy these criteria. A corollary is that certain "simple" Legendrian and transverse realizations of knots admitting grid number one diagrams maximize the classical invariants in their knot type. In order to obtain the above results, formulae are found for computing the classical invariants of Legendrian and transverse links from a toroidal front projection. Having these formulae, and known results about fibered links that support a given contact structure, it is found whether the duals of some families of Berge knots support the universally tight contact structure.
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