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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

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

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

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

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

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

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

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

Let M M be a cusped hyperbolic 3 3 -manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in P G L ( 2 , C ) \mathrm {PGL}(2,\mathbf {C}) (up to conjugation) is of complex dimension the number ν \nu of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the ν \nu peripheral tori). They indeed generically become non-discrete, except for a countable set. This last set corresponds to hyperbolic Dehn surgeries on M M , for which the peripheral representation is no more faithful. We work here in the framework of P G L ( n , C ) \mathrm {PGL}(n,\mathbf {C}) . The hyperbolic structure lifts, via the n n -dimensional irreducible representation, to a representation ρ g e o m \rho _{\mathrm {geom}} . We know from the work of Menal-Ferrer and Porti that the space of deformations of ρ geom \rho _{\textrm {geom}} has complex dimension ( n − 1 ) ν (n-1)\nu . We prove here that, unlike the P G L ( 2 ) \mathrm {PGL}(2) -case, the generic behaviour becomes the discreteness (and faithfulness) of the peripheral representation: in a neighbourhood of the geometric representation, the non-discrete peripheral representations are contained in a real analytic subvariety of codimension ≥ 1 \geq 1 .

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.

In the article we show that nondi eomorphic symplectic 4-manifolds whichadmit marked Lefschetz brations can share the same monodromy group. Explicitlywe prove that, for each integer g > 0, every knot surgery 4-manifold in a familyfE(2) K jK is a bered 2-bridge knot of genus g in S 3 g admits a marked Lefschetz bra-tion structure which has the same monodromy group. 1. IntroductionSeiberg-Witten invariants are one of the most powerful invariants in the classi- cation of smooth 4-manifolds and Fintushel-Stern’s knot surgery method is one ofthe most e ective methods to modify smooth structures on a given 4-manifold. ButSeiberg-Witten invariants are not complete invariants and there are known examplesof nondi eomorphic symplectic 4-manifolds which share the same Seiberg-Witteninvariants [3, 15].R. Fintushel and R. Stern showed that Seiberg-Witten invariants of knot surgery4-manifoldE(2) K = E(2)] F=m K S 1 (M K S 1 )can be computed by using the Alexander polynomial of the related knot K[2]. If werestrict our attention to a bered knot K, then E(2)

We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator 'A-hat' is a quantization of the knot complement's classical A-polynomial A(l,m). The construction proceeds by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in TQFT to put them back together. We advocate in particular that, properly interpreted, "gluing = symplectic reduction." We also arrive at a new finite-dimensional state integral model for computing the analytically continued "holomorphic blocks" that compose any physical Chern-Simons partition function., 110 pages, 14 figures; v2: references added and clarified, discussion of character varieties (Sec. 2) improved; v3: small typos corrected in Sec 6.3.2

Let G be a simply connected, simple, complex Lie group of rank 2. We give explicit Fock-Goncharov coordinates for configurations of triples and quadruples of affine flags in G. We show that the action on triples by orientation preserving permutations corresponds to explicit quiver mutations, and that the same holds for the flip (changing the diagonal in a quadrilateral). This gives explicit coordinates on higher Teichmuller space, and also coordinates for boundary-unipotent representations of 3-manifold groups. As an application, we compute the (generic) boundary-unipotent representations in Sp(4,C) for the figure-eight knot complement., 31 pages, 24 figures; Version 2 fixed several typos

We compute the Heegaard–Floer homology of \(S^{3}_{1}(K)\) (the (+1) surgery on the torus knot Tp,q) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath–Szabo d-invariant. We relate the result to known knot invariants of Tp,q as the genus and the Levine–Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard–Floer homologies of (+1) and (−1) surgeries on torus knots. This relation is best seen at the level of τ functions.

We show that one of the Cappell-Shaneson knot complements admits an extraordinarily small topological triangulation, containing only two four-dimensional simplices.

In this paper, we show the following result: Let K i be a knot in a closed orientable 3-manifold M i such that (M i ,K i ) is not homeomorphic to (S 2 ×S 1, x 0 ×S 1), i = 1, 2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E(K i ) is less than the difference of one and twice of the tunnel number of K i . Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive. We further show that if the distance of a Heegaard splitting of each knot complement is strictly bigger than twice the tunnel number of the knot (twice the sum of the tunnel number of the knot and one, respectively), then the tunnel number of connected sum of two such knots is additive (super additive, respectively).

In this paper, we define a new algebro-geometric invariant of three-manifolds resulting from Dehn surgery along a hyperbolic knot complement in S3. We establish a Casson-type invariant for these three-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications, as well as the computation of our new invariants for some three-manifolds resulting from Dehn surgery along the figure-eight knot.

A polynomial invariant depending on three variables is constructed for links in a thickened torus. The construction involves Kauffman’s formal knot theory based on the Dehn presentation of the knot group. Certain properties of the invariant are established, and a theorem about a Conway type relation is proved. Bibliography: 10 titles.

Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery on decayed knots produces surgery manifolds that have non-left-orderable fundamental group for all sufficiently positive surgeries. As an application, we prove that sufficiently positive cables of decayed knots are always decayed knots. These results mirror properties of L-space surgeries in the context of Heegaard Floer homology., 11 pages

We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms., 21 pages, 11 figures. Some errors in References are revised in version 2

In this paper we study submanifold with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if $K\subset (S^n, g)$ is a totally geodesic submanifold in a Riemannian sphere with positive sectional curvature where $n\ge 5$, then $K$ is homeomorphic to $S^{n-2}$ and the fundamental group of the knot complement $\pi_1(S^n-K)\cong \Bbb Z$., Comment: 9 pages

We construct a class of knots with the CI ∗ {}^* property, that is, π 1 ( M ( n ) ∣ ∂ M ( n ) ) ≠ { e } \pi _1(M(n)\mid \partial M(n))\neq \{e\} for some n > 0 n>0 . It follows that the infinite cyclic covering of such a knot cannot be embedded in any compact 3-manifold.

We show that every bridge surface of certain types of (1,1) prime knot has the disjoint curve property. Also we determine when a bridge surface of a pretzel knot of type (i2,3,n) has the disjoint curve property.

We give an estimation for the sharp-unknotting number of certain types of torus knots, and decide it for 39 torus knots.

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.

In the article, we study Fintushel-Stern's knot surgery four-manifold E(n) K and its monodromy factorization. For fibered knots we provide a smooth classification of knot surgery 4-manifolds up to twisted fiber sums. We then show that other constructions of 4-manifolds with the same Seiberg-Witten invariants are in fact diffeomorphic.

Let K be a knot in a closed orientable irreducible 3-manifold M and let P be a Heegaard splitting of the knot complement of genus at least two. Suppose Q is a bridge surface for K. Then either \begin{itemize} \item $d(P)\leq 2-\chi(Q-K)$, or \item K can be isotoped to be disjoint from Q so that after the isotopy Q is a Heegaard surface for the knot exterior and is isotopic to a possibly stabilized copy of P. \end{itemize}, Comment: References added, exposition slightly improved, 17 pages, 3 figures

Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot $K$ gives the Poincare homology sphere $\Sigma(2,3,5)$, then $K$ is the left-handed trefoil knot., Comment: The paper used to be known as "Knot Floer homology detects fibred links", however there was a mistake in the proof of Theorem 1.9 concerning links. The result about knots (Theorem 1.4) and the application to surgeries giving the Poincare' homology sphere (Corollary 1.7), as well as the applications by Ozsvath and Szabo (math.GT/0604079) and Yi Ni (math.GT/0607156) are uneffected by the mistake. This version is essentially the previous one, without Theorem 1.9 and Subsection 4.3. A more revised version is going to appear on Amer. J. Math. and is available on my personal webpage at http://www.labmath.uqam.ca/~ghiggini/papers.html

In this paper, we show that any non-arithmetic hyperbolic $2$-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic $2$-bridge link complement cannot irregularly cover a hyperbolic $3$-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of $3$-manifolds with non-trivial JSJ-decomposition and rank two fundamental groups. We also show that the only commensurable hyperbolic $2$-bridge link complements are the figure-eight knot complement and the $6_{2}^{2}$ link complement. Our work requires a careful analysis of the tilings of $\mathbb{R}^{2}$ that come from lifting the canonical triangulations of the cusps of hyperbolic $2$-bridge link complements., Comment: This is the final (accepted) version of this paper

We provide a new self contained proof of the following result of H. Schubert: If K is a (p,q)-torus knot, then the bridge number of K is min{p, q}.

It is known that the Brandt–Lickorish–Millett–Ho polynomialQcontains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable fromQis an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degreed≤ 10 are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special linear group of degree two over complex numbers. We also confirm the conjecture for the figure-eight knot and torus knots. This version is different from S. Gukov's because of a choice of polarization., 5 pages, added more comments

We define some new numerical invariants for knots with zero winding number in the solid torus. These invariants explore some geometric features of knots embedded in the solid torus. We characterize these invariants and interpret them on the level of the knot projection. We also find some relations among some of these invariants. Moreover, we give lower bounds for some of these invariants using Vassiliev invariants of type one. We connect our invariants to the bridge number in the solid torus. We give a lower bound and an upper bound of the wrap of a knot in the solid torus in terms of our new invariants.

This paper deals with the study of a new family of knot invariants: the L2-Alexander invariant. A main result is to give a method of computation of the L2-Alexander invariant of a knot complement using any presentation of default 1 of the knot group.

In this paper, in an attempt to extend an earlier work of Lück, we construct a knot invariant with parameter in C* by using the fundamental L2-representation of the fundamental group of the knot complement, which may be thought of as an L2-analogue of the usual Alexander polynomial of the knot in S3. When restricted to U(1) parameters, we interpret this invariant in terms of the U(1) twisted L2-Reidemeister torsion. We also show that this L2-invariant depends only on the norm |t| for t ∈ C*. In particular, this implies an unexpected rigidity property of the U(1) twisted L2-torsion on a knot complement. A possible relationship with the volume conjecture is discussed.

J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudo-Anosov maps have minimal bridge presentations which are thin., Comment: 14 pages, 4 figures

We write $K_1 \geq K_2$ for two prime knots $K_1,K_2$ if there exists a surjective group homomorphism from $G(K_1)$ onto $G(K_2)$ where $G(K_1), G(K_2)$ are the knot groups of $K_1,K_2$, respectively. In this paper, we determine this partial order for the knots in Rolfsen's knot table.

It is known that if any prime power branched cyclic cover of a knot in the 3-sphere is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in the 3-sphere whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of F_(1.0)/F_(1.5) for which Casson-Gordon invariants vanish in Cochran-Orr-Teichner's filtration of the classical knot concordance group . As a corollary, it follows that Casson-Gordon invariants are not a complete set of obstructions to a second layer of Whitney disks.

Abby Thompson proved that if a link $K$ is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin position but not bridge position then a thinnest level surface is essential. A theorem of Rieck and Sedgwick follows as a consequence, which says that thin position of a connected sum of small knots comes in the obvious way.

Let K be a knot in a closed orientable irreducible 3-manifold M. Suppose M admits a genus 1 Heegaard splitting and we denote by H the splitting torus. We say H is a 1-genus 1-bridge splitting of (M, K) if H intersects K transversely in two points, and divides (M, K) into two pairs of a solid torus and a boundary parallel arc in it. It is known that a 1-genus 1-bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If M = S 3 and the slope of the loop is longitudinal in one of the solid tori, then K is obtained by twisting a component of a 2-bridge link along the other component. We give a criterion for determining whether a given 1-genus 1-bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.

Novikov [12] initiated the study of the algebraic properties of quadratic forms over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom transversality construction of a Seifert surface of a knot and the infinite cyclic cover of the knot exterior. In this paper the analogy is applied to explain the relationship between the Seifert forms over a ring with involution A and Blanchfield forms over the Laurent polynomial extension A[z, z]. The rings A and A[z, z] correspond to the two ways of associating algebraic invariants to an n-knot k : S ⊂ S with A = Z : (i) The Z[z, z]-module invariants of the canonical infinite cyclic cover M = pR of the exterior of k M = cl.(S\k(S)×D) ⊂ S with k(S) × D ⊂ S a regular neighbourhood of k(S) in S, p : M → S a map inducing an isomorphism p : H(S) ∼= H(M), and ∂M = S × S. (ii) The Z-module invariants of a codimension 1 submanifold N ⊂ S with boundary ∂N = k(S) ⊂ S

We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yields an L-space knot. Consequently, any knot obtained by a non-trivial band sum cannot admit a finite surgery. For context, we exhibit two examples of non-trivial band sums of tight fibered knots producing prime knots: one is fibered but not tight, and the other is strongly quasipositive but not fibered., Comment: 8 pages, 6 figures

We study the existence of relations between the degrees of the knot polynomials and some classical knot invariants, partially confirming and extending the question of Morton on the skein polynomial and a recent question of Ferrand.

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a noncommutative ring from H-triangulations of 3-manifolds and observed that for trefoil and figure-eight knot complements the abelianization of this ring is isomorphic to the ring of regular functions on the deformation variety, Kashaev, [Formula: see text]-groupoids in knot theory, Geom. Dedicata 150(1) (2010) 105–130; Kashaev, Noncommutative teichmüller spaces and deformation varieties of knot completeness; Kashaev, Delta-groupoids and ideal triangulation in Chern–Simons gauge theory: 20 Years After, AMS/IP Studies Advanced Mathematics, Vol. 50 (American Mathematical Society, RI, 2011). In this paper, we prove that this is true for any knot complement in a homology sphere. We also analyze some examples on other manifolds.

The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribbon-graph polynomial invariants., 18 pages

There is a positive constant c 1 c_1 such that for any diagram D \mathcal {D} representing the unknot, there is a sequence of at most 2 c 1 n 2^{c_1 n} Reidemeister moves that will convert it to a trivial knot diagram, where n n is the number of crossings in D \mathcal {D} . A similar result holds for elementary moves on a polygonal knot K K embedded in the 1-skeleton of the interior of a compact, orientable, triangulated P L PL 3-manifold M M . There is a positive constant c 2 c_2 such that for each t ≥ 1 t \geq 1 , if M M consists of t t tetrahedra and K K is unknotted, then there is a sequence of at most 2 c 2 t 2^{c_2 t} elementary moves in M M which transforms K K to a triangle contained inside one tetrahedron of M M . We obtain explicit values for c 1 c_1 and c 2 c_2 .

We establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K ⊂ S3 be slice. We produce a new family of metabelian obstructions to slicing K such as those first developed by Casson and Gordon in [4] in the mid 1970s. Surgery is used to turn the knot complement S3 − K into a closed manifold M and, for given unitary representations of π1(M), η can be defined. Levine has recently shown in [11] that η acts as an homology cobordism invariant for a certain subvariety of the representation space of π1(N), where N is zero-framed surgery on a knot concordance. We demonstrate a large family of such representations, show they are extensions of similar representations on the boundary of N and prove that for slice knots, the value of η defined by these representations must vanish.The paper is organized as follows; Section 1 consists of background material on η and Levine's work on how it is used as a concordance invariant [11]. Section 2 deals with unitary representations of π1(M) and is broken into two parts. In 2·1, homomorphisms from π1(M) to a metabelian group Γ are developed using the Blanchfield pairing. Unitary representations of Γ are then considered in 2·2. Conditions ensuring that such two stage representations of π1(M) allow η to be used as an invariant are developed in Section 3 and [Pscr ]k, the family of such representations, is defined. Section 4 contains the main result of the paper, Theorem 4·3. Lastly, in Section 5, we demonstrate the construction of representations in [Pscr ]k.