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A knot $\kappa$ in $S^3$ is persistently foliar if, for each non-trivial boundary slope, there is a co-oriented taut foliation meeting the boundary of the knot complement transversely in a foliation by curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a co-oriented taut foliation in every manifold obtained by non-trivial Dehn surgery on that knot. We show that any composite knot with a persistently foliar summand is persistently foliar and that any nontrivial connected sum of fibered knots is persistently foliar. As an application, it follows that any composite knot in which each of two summands is fibered or at least one summand is nontorus alternating or Montesinos is persistently foliar. We note that, in constructing foliations in the complements of fibered summands, we build branched surfaces whose complementary regions agree with those of Gabai's product disk decompositions, except for the one containing the boundary of the knot complement. It is this boundary region which provides for persistence., Comment: 37 pages, 30 figures. Added description of canonical meridian and two new figures. Improved exposition and stronger statement of results. To be published in Algebraic and Geometric Topology

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds $M_3$. In particular, this directly indicates that the homological block for the torus knot complement in $S^3$ is an analytic continuation of the full $G=SU(2)$ partition function, i.e. the colored Jones polynomial., Comment: 42 pages, 4 figures

This is a sequel to the authors' article [BKO](arXiv:1901.02239). We consider a hyperbolic knot $K$ in a closed 3-manifold $M$ and the cotangent bundle of its complement $M \setminus K$. We equip $M \setminus K$ with a hyperbolic metric $h$ and its cotangent bundle $T^*(M \setminus K)$ with the induced kinetic energy Hamiltonian $H_h = \frac{1}{2} |p|_h^2$ and Sasakian almost complex structure $J_h$, and associate a wrapped Fukaya category to $T^*(M\setminus K)$ whose wrapping is given by $H_h$. We then consider the conormal $\nu^*T$ of a horo-torus $T$ as its object. We prove that all non-constant Hamiltonian chords are transversal and of Morse index 0 relative to the horo-torus $T$, and so that the structure maps satisfy $\widetilde{\mathfrak m}^k = 0$ unless $k \neq 2$ and an $A_\infty$-algebra associated to $\nu^*T$ is reduced to a noncommutative algebra concentrated to degree 0. We prove that the wrapped Floer cohomology $HW(\nu^*T; H_h)$ with respect to $H_h$ is well-defined and isomorphic to the Knot Floer cohomology $HW(\partial_\infty(M \setminus K))$ that was introduced in [BKO] for arbitrary knot $K \subset M$. We also define a reduced cohomology, denoted by $\widetilde{HW}^d(\partial_\infty(M \setminus K))$, by modding out constant chords and prove that if $\widetilde{HW}^d(\partial_\infty(M \setminus K))\neq 0$ for some $d \geq 1$, then $K$ cannot be hyperbolic. On the other hand, we prove that all torus knots have $\widetilde{HW}^1(\partial_\infty(M \setminus K)) \neq 0$., Comment: 52 pages, 1 figure; v2 57 pages, 2 figures, calculations for torus knots added, abstract and introduction rewritten, mistakes in the statements and proofs of Proposition 2.1 and Lemma 9.4 corrected, old Section 11 moved to Appendix B, typos corrected

We study notions of complexity for link complement states in Chern-Simons theory with compact gauge group G. Such states are obtained by the Euclidean path integral on the complement of n-component links inside a 3-manifold M3. For the Abelian theory at level k we find that a natural set of fundamental gates exists, and one can identify the complexity as differences of linking numbers modulo k. Such linking numbers can be viewed as coordinates which embeds all link complement states into Zk⊗n(n−1)/2, and the complexity is identified as the distance with respect to a particular norm. For non-Abelian Chern-Simons theories, the situation is much more complicated. We focus here on torus link states and show that the problem can be reduced to defining complexity for a single knot complement state. We suggest a systematic way to choose a set of minimal universal generators for single knot complement states and then evaluate the complexity using such generators. A detailed illustration is shown for SU(2)k Chern-Simons theory, and the results can be extended to a general compact gauge group.

Let $K$ be a knot in an L-space $Y$ with a Dehn surgery to a surface bundle over $S^1$. We prove that $K$ is rationally fibered, that is, the knot complement admits a fibration over $S^1$. As part of the proof, we show that if $K\subset Y$ has a Dehn surgery to $S^1 \times S^2$, then $K$ is rationally fibered. In the case that $K$ admits some $S^1 \times S^2$ surgery, $K$ is Floer simple, that is, the rank of $\hat{HFK}(Y,K)$ is equal to the order of $H_1(Y)$. By combining the latter two facts, we deduce that the induced contact structure on the ambient manifold $Y$ is tight. In a different direction, we show that if $K$ is a knot in an L-space $Y$, then any Thurston norm minimizing rational Seifert surface for $K$ extends to a Thurston norm minimizing surface in the manifold obtained by the null surgery on $K$ (i.e., the unique surgery on $K$ with $b_1>0$)., 25 pages, 1 figure; v2: minor revisions throughout. This is the version to appear in Transactions of the AMS

Tied links in $S^3$ were introduced by Aicardi and Juyumaya as standard links in $S^3$ equipped with some non-embedded arcs, called {\it ties}, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces $L(p,1)$, in handlebodies of genus $g$, and in the complement of the $g$-component unlink. We introduce the tied braid groups $TB_{g, n}$ by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we state and prove Alexander's and Markov's theorems for tied links in the 3-manifolds mentioned above. Finally, we emphasize on further steps needed in order to study tied links in knot complements and c.c.o. 3-manifolds, which is the subject of a sequel paper., Comment: 20 pages, 25 figures

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston's geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al., Comment: 18 pages, 5 Figure, Special Issue "Groups, Geometry and Topology for Quantum Computations" in Quantum Reports

The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the R\'enyi entropy of index $m$, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with $S^3$ complements of a two-component link which is a connected sum of a knot $\mathcal{K}$ and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the $m$-moment of the reduced density matrix as a three-manifold invariant $Z(M_{\mathcal{K}_m})$, which is the partition function of $M_{\mathcal{K}_m}$. Here $M_{\mathcal{K}_m}$ is a closed 3-manifold associated with the knot $\mathcal K_m$, where $\mathcal K_m$ is a connected sum of $m$-copies of $\mathcal{K}$ (i.e., $\mathcal{K}\#\mathcal{K}\ldots\#\mathcal{K}$) which mimics the well-known replica method. We analyse the partition functions $Z(M_{\mathcal{K}_m})$ for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling $k$. For SU(2) group, we show that $Z(M_{\mathcal{K}_m})$ can grow at most polynomially in $k$. On the contrary, we conjecture that $Z(M_{\mathcal{K}_m})$ for SO(3) group shows an exponential growth in $k$, where the leading term of $\ln Z(M_{\mathcal{K}_m})$ is the hyperbolic volume of the knot complement $S^3\backslash \mathcal{K}_m$. We further propose that the R\'enyi entropies associated with SO(3) group converge to a finite value in the large $k$ limit. We present some examples to validate our conjecture and proposal., Comment: 38 pages, 24 figures & 15 tables; v2: Introduction & Conclusion modified, new subsection added in section 3, three new references added; matches published version

In this paper we compute the Kauffman bracket skein module of the complement of $(2, 2p+1)$-torus knots, $KBSM(T_{(2, 2p+1)}^c)$, via braids. We start by considering geometric mixed braids in $S^3$, the closure of which are mixed links in $S^3$ that represent links in the complement of $(2, 2p+1)$-torus knots, $T_{(2, 2p+1)}^c$. Using the technique of parting and combing, we obtain algebraic mixed braids, that is, mixed braids that belong to the mixed braid group $B_{2, n}$ and that are followed by their ``coset'' part, that represents $T_{(2, 2p+1)}^c$. In that way we show that links in $T_{(2, 2p+1)}^c$ may be pushed to the genus 2 handlebody, $H_2$, and we establish a relation between $KBSM(T_{(2, 2p+1)}^c)$ and $KBSM(H_2)$. In particular, we show that in order to compute $KBSM(T_{(2, 2p+1)}^c)$ it suffices to consider a basis of $KBSM(H_2)$ and study the effect of combing on elements in this basis. We consider the standard basis of $KBSM(H_2)$ and we show how to treat its elements in $KBSM(T_{(2, 2p+1)}^c)$, passing through many different spanning sets for $KBSM(T_{(2, 2p+1)}^c)$. These spanning sets form the intermediate steps in order to reach at the set $\mathcal{B}_{T_{(2, 2p+1)}^c}$, which, using an ordering relation and the notion of total winding, we prove that it forms a basis for $KBSM(T_{(2, 2p+1)}^c)$. We finally consider c.c.o. 3-manifolds $M$ obtained from $S^3$ by surgery along the trefoil knot and we discuss steps needed in order to compute the Kauffman bracket skein module of $M$. We first demonstrate the process described before for computing the Kauffman bracket skein module of the complement of the trefoil, $KBSM(Tr^c)$, and we study the effect of braid band moves on elements in the basis of $KBSM(Tr^c)$. These moves reflect isotopy in $M$ and are similar to the second Kirby moves., Comment: 27 pages, 18 figures

The Gukov-Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of `strange identities' for positive braid knots., 27 pages, 13 figures. v2 minor corrections

A relation between the two-variable series knot invariant and the Akutus-Deguchi-Ohtsuki(ADO)-invariant was conjectured recently. We reinforce the conjecture by presenting explicit formulas and/or an algorithm for certain ADO-invariants of torus knots obtained from the series invariant of complement of a knot. Furthermore, one parameter deformation of ADO_3-polynomial of torus knots is provided., The published version of the paper

The waist size of a cusp in an orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In this paper, it is proved that the next two smallest waist sizes are realized uniquely for the cusps in the $5_2$ knot complement and the manifold obtained by (2,1)-surgery on the Whitehead link. One application is an improvement on the universal upper bound for the length of an unknotting tunnel in a 2-cusped hyperbolic 3-manifold., Comment: 15 pages, 10 figures

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3)., 13 pages, 25 figures

It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirtinger generators for the representation and also cusp shape. We demonstrate explicit solutions for $T(2,N)$ torus knots, $J(N,M)$ knots and also for other interesting knots as examples. Using these solutions we can observe the asymptotic behavior of complex volumes and cusp shapes of these knots. We note that this construction works for any knot or link, and reflects systematically both geometric properties of the knot complement and combinatorial aspect of the knot diagram., 55 pages, 31 figures

We construct infinitely many examples of pairs of isospectral but non-isometric [Formula: see text]-cusped hyperbolic [Formula: see text]-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada’s method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.

We review some recent developments in Chern-Simons theory on a hyperbolic 3-manifold $M$ with complex gauge group $G$. We focus on the case $G=SL(N,\mathbb{C})$ and with $M$ a knot complement. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral. We also review various applications and open questions regarding the cluster partition function and some of its relation with string theory., Comment: 26 pages, 1 figure, contribution to the proceedings of the workshop "Non-abelian Gauged Linear Sigma Model and Geometric Representation Theory" held at BICMR (Jun. 2015). To appear in a special issue of Chinese Annals of Mathematics, Series B; v2: reference added

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $\hat{A}$ polynomial), with a classical invariant, namely the defining polynomial $A$ of the $\psl$ character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the $\hat{A}$-polynomial (after we set $q=1$, and excluding those of $L$-degree zero) coincides with those of the $A$-polynomial. In this paper, we introduce a version of the $\hat{A}$-polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the $\hat{A}$-polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the $R$-matrix state sum formula for the colored Jones polynomial, and its certificate., Comment: 34 pages, 12 figures

An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, $\text{Vol}(K)$. A less studied question is whether $\text{Vol}(K)$ can be recovered directly from the original Jones polynomial ($N = 2$). In this report we use a deep neural network to approximate $\text{Vol}(K)$ from the Jones polynomial. Our network is robust and correctly predicts the volume with $97.6\%$ accuracy when training on $10\%$ of the data. This points to the existence of a more direct connection between the hyperbolic volume and the Jones polynomial., 18 pages, 9 figures, updated figures

When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with non-homeomorphic complements., 17 pages, 4 figures, 1 table

We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-manifold. This the first example of geometrically bounding hyperbolic knot complement and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume., 9 pages, 4 figures, typos corrected, improved exposition of tetrahedral manifolds. Added Proposition 3.3, which gives necessary and sufficient conditions for M_T to be a manifold, and Remark 4.4, which shows that the figure-eight knot bounds a 4-manifold of minimal volume. Updated bibliography

We introduce a knot semigroup as a cancellative semigroup whose defining relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we define it is closely related to such tools of knot theory as the twofold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T(2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally defined factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research.

In this chapter, we construct some examples of hyperbolic 3-manifolds. We begin with a geometric method for constructing spherical, Euclidean, and hyperbolic 3-manifolds in Sections 10.1 and 10.2. Examples of complete hyperbolic 3-manifolds of finite volume are constructed in Section 10.3. The problem of computing the volume of a hyperbolic 3-manifold is taken up in Section 10.4. The chapter ends with a detailed study of hyperbolic Dehn surgery on the figure-eight knot complement.

Dehn surgery and the notion of reducible manifolds are both important tools in the study of 3-manifolds. The Cabling Conjecture of Francisco Gonzalez-Acuna and Hamish Short describes the purported circumstances under which Dehn surgery can produce a reducible manifold. This thesis extends the work of James Allen Hoffman, who proved the Cabling Conjecture for knots of bridge number up to four. Hoffman built upon the combinatorial machinery used by Cameron Gordon and John Luecke in their solution to the knot complement problem. The combinatorial approach starts with the graphs of intersection of a thin level sphere of the knot and the reducing sphere in the surgered manifold. Gordon and Luecke’s proof then proceeds by induction on certain cycles. Hoffman provides more insight into the structure of the base case of the induction (i.e. in an innermost cycle or a graph containing no such cycles). Hoffman uses this structure in a case-by-case proof of the Cabling Conjecture for knots of bridge number up to four. We find trees with specific properties in the graph of intersection, and use them to prove the existence of structure which provides lower bounds on the number of the aforementioned innermost cycles. Our results combined with a recent lower bound on the number of vertices inside the innermost cycles succinctly prove the conjecture for bridge number up to five and suggests an approach to the conjecture for knots of higher bridge number.

The traditional skein relation for the Alexander polynomial involves an oriented knot, K + , with a distinguished positive crossing; a knot K − , obtained by changing the distinguished positive crossing of K + to a negative crossing; and a link K 0 , the orientation preserving resolution of the distinguished crossing. We refer to ( K + , K − , K 0 ) as the oriented skein triple . A tangle is defined as a pair ( B, t ) of a 3-dimensional ball B and a collection of disjoint, simple, properly embedded arcs, denoted t . DeWitt Sumners and Claus Ernst developed the tangle model which uses the mathematics of tangles to model DNA-protein binding. The protein is seen as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Effectively, the action of these proteins can be modeled as K − ↔ K + . Recombinases are proteins that cut two segments of DNA and recombine them in some manner. While recombinase local action varies, most are mathematically equivalent to a resolution, i.e. K ± ↔ K 0 . The oriented triple is now viewed as K − = circular DNA substrate, K + = product of topoisomerase action, K 0 = product of recombinase action. The theorem stated in this dissertation gives a relationship between two 2-bridge knots, K + and K − , that differ by a crossing change and a link, K 0 created from the oriented resolution of that crossing. We apply this theorem to difference topology experiments using topoisomerase proteins to study SMC proteins. In recent years, link homology theories have become a popular invariant to develop and study. One such invariant knot Floer homology , was constructed by Peter Ozsvath, Zoltan Szabo, and independently Jacob Rasmussen, denoted by HFK . It is also a refinement of a classical invariant, the Alexander polynomial . The study of DNA knots and links are of great interest to molecular biologists as they are present in many cellular process. The variety of experimentally observed DNA knots and links makes separating and categorizing these molecules a critical issue. Thus, knowing the knot Floer homology will provide restrictions on knotted and linked products of protein action. We give a summary of the combinatorial version of knot Floer homology from known work, providing a worked out example. The thesis ends with reviewing knot Floer homology properties of three particular sub-families of biologically relevant links known as (2, p)- torus links, clasp knots and 3-strand pretzel…

Since its inception, the notion of thin position has played an important role in low-dimensional topology. Thin position for knots in the 3-sphere was first introduced by David Gabai in order to prove the Property R Conjecture. In addition, this theory factored into Cameron Gordon and John Luecke’s proof of the knot complement problem and revolutionized the study of Heegaard splittings upon its adaptation by Martin Scharlemann and Abigail Thompson. Let h : S → R be a Morse function with two critical points. Loosely, thin position of a knot K in S is a particular embedding of K which minimizes the total number of intersections with a maximal collection of regular level sets, where this number of intersections is called the width of the knot. Although not immediately obvious, it has been demonstrated that there is a close relationship between a thin position of a knot K and essential meridional planar surfaces in its exterior E(K). In this thesis, we study the nature of thin position under knot companionship; namely, for several families of knots we establish a lower bound for the width of a satellite knot based on the width of its companion and the wrapping or winding number of its pattern. For one such class of knots, cable knots, in addition to finding thin position for these knots, we establish a criterion under which non-minimal bridge positions of cable knots are stabilized. Finally, we exhibit an embedding of the unknot whose width must be increased before it can be simplified to thin position.

Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic $3$-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of non-elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings., Comment: Minor modification (including typesetting). 56 pages

Due to the Hyperbolization Theorem, we know precisely when does a given compact three dimensional manifold admits a hyperbolic metric. Moreover, by the Mostow's Rigidity Theorem this geometric structure is unique. However, finding effective and computable connections between the geometry and topology is a challenging problem. Most of the results on this thesis fit into the theme of making the connections more concrete. To every oriented closed geodesic on a hyperbolic surface has a canonical lift on the unit tangent bundle of the surface, and we can see it as a knot in a three dimensional manifold. The knot complement given in this way has a hyperbolic structure. The objective of this thesis is to estimate the volume of the canonical lift complement. For every hyperbolic surface we give a sequence of geodesics on the surface, such that the knot complements associated are not homeomorphic with each other and the sequence of the corresponding volumes is bounded. We also give a lower bound of the volume of the canonical lift complement by an explicit real number which describes a relation between the geodesic and a pants decomposition of the surface. This give us a method to construct a sequence of geodesics where the volume of the associated knot complements is bounded from below in terms of the length of the corresponding geodesic. For the particular case of the modular surface, we obtain estimations for the volume of the canonical lift complement in terms of the period of the continuous fraction expansion of the corresponding geodesic.; Grâce au théorème d'hyperbolisation, nous savons précisément quand une variété de dimension trois compacte admet une métrique hyperbolique. Par ailleurs, d'après le théorème de rigidité de Mostow, cette structure géométrique est unique. Cependant, trouver des liens pratiques entre la géométrie et la topologie est un problème difficile. La plupart des résultats décrits dans cette thèse visent à concrétiser ces liens. Toute géodésique fermée orientée dans une surface hyperbolique admet un relèvement canonique dans le fibré tangent unitaire de la surface, et on peut donc le voir comme un nœud dans une variété de dimension trois. Les extérieurs des nœuds ainsi construits admettent une structure hyperbolique. Cette thèse a pour objet d'estimer le volume des extérieurs des relèvements canoniques. Pour toute surface hyperbolique on construit une suite de géodésique sur la surface, tel que les extérieurs associées ne sont pas homéomorphes entre elles et dont la suite des volumes respectifs est bornée. Aussi on minore le volume de l'extérieur à l'aide d'un réel explicite qui décrit une relation entre la géodésique et une décomposition en pantalons de la surface. Ceci donne une méthode pour construire une suite de géodésiques dont les volumes des extérieurs associées sont minorées en termes de la longueur de la géodésique correspondant. Dans le cas particulier de la surface modulaire, on obtient des estimations du volume de l'extérieur en termes de la période de la fraction continue associée à la géodésique.

We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). For hyperbolic integer homology spheres this comes with the definition, and for Seifert fibered integer homology spheres this is well known. We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation. By work of Boileau, Rubinstein, and Wang, the general case follows. Using a result of Kuperberg, we get the corollary that the problem of 3-sphere recognition is in the complexity class coNP, provided the generalised Riemann hypothesis holds. To prove our result, we establish a topological fact about the image of the SU(2)-representation variety of a non-trivial knot complement into the representation variety of its boundary torus, a pillowcase. For this, we use holonomy perturbations of the Chern-Simons function in an exhaustive way - we show that any area-preserving self-map of the pillowcase fixing the four singular points, and which is isotopic to the identity, can be C^0-approximated by maps which are realised geometrically through holonomy perturbations of the flatness equation in a thickened torus. To conclude, we use a stretching argument in instanton gauge theory, and a non-vanishing result of Kronheimer and Mrowka for Donaldson's invariants of a 4-manifold which contains the 0-surgery of a knot as a splitting hypersurface., Comment: 58 pages, 2 figures; v2: a few typos corrected; v3: results of section 10 strengthened, results about the complexity of 3-recognition added, outline of the technical main result added to the introduction, updated acknowledgements and references; v4: revision after two referee reports, to appear in Duke Math. J

This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective computations. The talk was given in the Mathematische Arbeitstagung June 24-July 1, 2011. Updated the bibliography., Comment: 17 pages, 13 figures, Arbeitstagung talk Bonn 2011

A twisted torus knot is obtained from a torus knot by adding a number of full twists to some adjacent strands of the torus knot. In this paper, we show that if a twisted torus knot is a torus knot, then the number of added full twists is generically at most two in absolute value. We also show that this bound is the best possible by classifying twisted torus knots for which the upper bound is attained. 57N10

A knot shadow is a diagram with all crossing information missing. We cannot determine the original knot from a knot shadow in general. In this paper, we investigate properties (unknotting number, genus, braid index, etc.) of the original knot from a knot shadow.

It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement. In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot. 57M27

In 1982 Louis Kauffman conjectured that if a knot in the 3-sphere is a slice knot then on any Seifert surface for that knot there exists a homologically essential simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explictly expressed in terms of invariants of such curves on the Seifert surface. We give counterexamples to Kauffman's conjecture, that is, we exhibit (smoothly) slice knots that admit (unique minimal genus) Seifert surfaces on which every homologically essential simple closed curve of self-linking zero has non-zero Arf invariant and non-zero signatures., 17 pages, 11 Figures, minor corrections/clarifications and up-dated references in version 2

In this chapter, we describe representations of the fundamental group of a knot complement to \(\mathrm {SL}(2;\mathbb {C})\) by giving examples. We also give the definitions of the Chern–Simons invariant and the Reidemeister torsion associated with such a representation. We also give examples of calculation. We will explain relations of these invariants to the asymptotic behavior of the colored Jones polynomial in the next chapter.

In this chapter, for a hyperbolic knot K, we explain an idea of a possible proof of the Volume Conjecture by using Kashaev’s invariant 〈K〉N of K, which is known to be the N-colored Jones polynomial JN(K, q) evaluated at $$\displaystyle q=\exp {2\pi \sqrt {-1}\over N}$$ after the work of Murakami and Murakami (Acta Math 186(1):85–104, 2001. MR 1828373). By using 〈K〉N rather than JN(K, q), we can observe the correspondence between the algebraic structure of 〈K〉N and the geometric structure of the complement of K more clearly. Throughout this chapter, we set q as above. In the first section, following Yokota (Interdiscip Inf Sci 9(1):11–21, 2003. MR MR2023102 (2004j:57014)), we explain how to compute the invariant and how to reduce it. In the second section, following Kashaev and Yokota (On the volume conjecture for 52, Preprint, 2012), we explain how to compute the asymptotic behavior of an integral expression of the invariant. In the third section, following Yokota (Interdiscip Inf Sci 9(1):11–21, 2003. MR MR2023102 (2004j:57014)) again, we explain the relationship between the hyperbolic structure of the knot complement, and a “potential” function which we obtain in the second section. In the fourth section, we sort the remaining tasks.

In this chapter we show various generalizations of the volume conjecture. Firstly, we introduce the complexification of the conjecture by studying the imaginary part of \(\log {J_N(K;\exp (2\pi \sqrt {-1}/N))}\). We expect the (\(\mathrm {SL}(2;\mathbb {C})\)) Chern–Simons invariant to appear. Secondly, we refine the conjecture by considering more precise approximation of the colored Jones polynomial. We conjecture that the Reidemeister torsion would appear. Lastly, we perturb \(2\pi \sqrt {-1}\) in \(\exp (2\pi \sqrt {-1}/N)\) slightly and see what happens to the asymptotic expansion of the colored Jones polynomial. The corresponding topological phenomenon is to perturb the hyperbolic structure of the knot complement, provided the knot is hyperbolic. If the knot is non-hyperbolic we expect various representations of the fundamental group of the knot complement to \(\mathrm {SL}(2;\mathbb {C})\).

It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3-manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold., 10 pages, 9 figures

We present a new paradigm for three dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension three is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (1) For certain parameters, the Lorenz system has an invariant one dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (2) The flow is topologically equivalent to an Anosov flow on the complement of this curve, and even to a geodesic flow. (3) When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement, and different knots appear for different parameter values. The steps of a mathematical proof of these statements are at different stages. Some have been proven, for some we present numerical evidence and some are still conjectural., 14 pages, 11 figures

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot [Formula: see text]-mosaic is an [Formula: see text] matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot [Formula: see text]-mosaics for any positive integers [Formula: see text] and [Formula: see text], toroidal knot [Formula: see text]-mosaics for co-prime integers [Formula: see text] and [Formula: see text], and furthermore toroidal knot [Formula: see text]-mosaics for a prime number [Formula: see text]. We also analyze the asymptotics of the growth rates of their cardinality.