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We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjecture can be restated as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjecture is true, then we can prove that a knot is trivial if and only if all of its Vassiliev invariants are trivial., 18 pages. Added Remark 2.2 and improved Remark 5.7

Let M be a cusped hyperbolic 3-manifold containing an incompressible thrice punctured sphere S. Suppose that M is not the Whitehead link complement. We prove that a certain arc on S is isotopic to an edge of a Euclidean decomposition of M. By using the above result, we relate alternating knot diagrams and the canonical decompositions. Let K be an alternating hyperbolic knot. On a reduced alternating knot diagram of K, if we replace one of the crossings with a large number of half twists, the polar axis of the crossing is isotopic to an edge of the canonical decomposition for the resulting knot.

Complex superpositions of degenerate hydrogen wavefunctions for the n th energy level can possess zero lines (phase singularities) in the form of knots and links. A recipe is given for constructing any torus knot. The simplest cases are constructed explicitly: the elementary link, requiring n≥6, and the trefoil knot, requiring n≥7. The knots are threaded by multistranded twisted chains of zeros. Some speculations about knots in general complex quantum energy eigenfunctions are presented.

We consider Legendrian knots and links in the standard 3-dimensional contact space. In 1997 Chekanov [Ch] introduced a new invariant for these knots. At the same time, a similar construction was suggested by Eliashberg [E1] within the framework of his joing work with Hofer and Givernthal on Symplectic Field Theory ([E2],[EGH]). To a knot diagram, they associated a differential algebra A. Its stable isomorphism type is invariant under Legendrian isotopy of the knot. In this paper, we introduce an additional structure on this algebra in the case of a Legendrian link. For a link of N components, we show that its algebra splits A = ⊕g ∈ G Ag Here G is a free group on (N - 1) variables. The splitting is determined by the order of the knots and is preserved by the differential. It gives a tool to show that some permutations of link components are impossible to produce by Legendrian isotopy.

In this paper, we shall give a general formula for the quantum SU(2) -invariant of Seifert fibered rational homology spheres over a 2-sphere associated with the trivial first cohomology class modulo two, based on the linear skein theory. We shall derive the Casson–Walker invariant from the formula. Also, we shall give a formula for a second coefficient of the perturbative invariant associated with the trivial first cohomology class modulo two, which is a topological invariant. We shall give a relation between the second coefficient and the Ohtsuki invariant. We shall show that the second coefficient for Seifert fibered integral homology spheres belongs to 3 Z .

Virtual knots, defined by Kauffman, provide a natural generalization of classical knots. Most invariants of knots extend in a natural way to give invariants of virtual knots. In this paper we study the fundamental groups of virtual knots and observe several new and unexpected phenomena. In the classical setting, if the longitude of a knot is trivial in the knot group then the group is infinite cyclic. We show that for any classical knot group there is a virtual knot with that group and trivial longitude. It is well known that the second homology of a classical knot group is trivial. We provide counterexamples of this for virtual knots. For an arbitrary group G, we give necessary and sufficient conditions for the existence of a virtual knot group that maps onto G with specified behavior on the peripheral subgroup. These conditions simplify those that arise in the classical setting.

We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to classical knots is an invariant of degree ⩽n in the classical sense. A universal invariant of degree ⩽n is defined via a Gauss diagram formula. This machinery is used to obtain explicit formulas for invariants of low degrees. The same technique is also used to prove that any finite type invariant of classical knots is given by a Gauss diagram formula. We introduce the notion of n-equivalence of Gauss diagrams and announce virtual counter-parts of results concerning classical n-equivalence.

A smooth knot is non-trivial precisely if it cannot be unlinked from every small deformation of itself. White's formula expresses the linking number of a representative of a knot and a deformation of it as a sum of two, not necessarily integral, terms — a writhe and a twist. An elementary proof is given of this formula. A byproduct of the proof is an easy derivation of the expression for the writhe of the knot in terms of standard invariants of the curve and the writhe of the knot diagram obtained by projection onto a plane.

The notion of an abstract link diagram is re-introduced with a relationship with Kauffman's virtual knot theory. It is prove that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams. Using abstract link diagrams, we have a geometric interpretation of the group and the quandle of a virtual knot. A generalization to higher dimensional cases is introduced, and the state-sum invariants are treated.

Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss codes via knot diagrams with virtual crossings. This paper gives basic results and examples (such as non-trivial virtual knots with trivial Jones polynomial), studies fundamental group and quandles of virtual knots, extensions of the bracket and Jones polynomials, quantum link invariants with virtual framings, Vassiliev invariants and applications to knots in thickened surfaces., Comment: LaTeX, 53 pages, 25 figures

Mackays generalization of Penrose tiling is shown to be related to the Hirzbruchsignature, δ, of four manifolds in the case of a Murray–von Neumann continuousdimension in E (∞) space. It is further shown that σ is numerically identicalto the Jones knot invariant, VL, for the right-handed trefoil. Finally, aconjecture regarding the theorem of local isomorphism is presented.

We introduce the beta function of a knot in euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues., Comment: plain TeX; the new version discusses the relation with the M\"obius energy for knots

An ideal point is a limit of conjugacy classes of representations of the fundamental group Γ of a 3-manifold into SL 2 ( C ) . Culler and Shalen (1983) studied ideal points by using trees. In this paper we give a method to construct ideal points by making the trees concretely for knot groups Γ. We explain the method by illustrating it for simple three knots.

We prove that there is no non-constant Vassiliev invariant which is constant on alternating knots of Infinitely many genera (contrasting the existence of the Conway Vassiliev invariants, which vanish on any finite set of genera) and that a (non-constant) knot invariant with values bounded by a funciton of the genus, in particular any invariant depending just on genus, signature and maximal degree of the Alexander polynomial, is not a Vassiliev invariant.

An alternative formulation is proposed for the calculation of the non-adiabatic non-Abelian connection and phase factor by making use of the invariant theory and invariant-related unitary transformation method. In order to illustrate the formulation, we study the non-adiabatic non-Abelian phase associated with the motion of a charged particle in a time-dependent magnetic field (the time-dependent Landau level problem).

Based on certain analogies between Jones knot invariant and KAM theorem, it is conjectured that a generic D3 nonlinear dynamical set will evolve towards a state corresponding to a four dimensional central manifold M4 with signature τ equal to the inverse of the index [M:N]=1/φ3, where 1/φ3 is the Subtle Mean and φ=( 5 −1)/2 is the Golden Mean. It is further conjectured that this selfsimilar transfinite four manifolds marks the critical limit for the onset of global chaos.

We show that any Heegaard splitting of the trivial knot in a compact orientable 3-manifold is standard.

We give an algorithm for calculating the second degree coefficient of the Conway polynomial of a ribbon 1-knot. This naturally yields a recursive calculation for the second derivative at t = 1, Δ′′(1), of the normalized Alexander polynomial of a ribbon 2-knot in R4, which is the first nontrivial finite type invariant of a ribbon 2-knot defined by Habiro, Kanenobu, and Shima.

Let M be the manifold obtained by 0-framed surgery along a knot K in the 3-sphere. A Topological Quantum Field Theory assigns to a fundamental domain of the universal abelian cover of M an operator, whose non-nilpotent part is the Turaev-Viro module of K. In this paper, using surgery formulas, we give a matrix presentation for the Turaev-Viro module of any knot K, in the case of the (Vp, Zp) TQFT of Blanchet, Habegger, Masbaum and Vogel. We do the computation for a family of knots in the special case p = 8, and note the relation with the fibering question.

The first part of the paper attempts to establish connections between hypersphere backing in infinite dimensions, the expectation value of dim E (∞) spacetime and the COBE measurement of the microwave background radiation. One of the main results reported here is that the mean sphere in S (∞) spans a four dimensional manifold and is thus equal to the expectation value of the topological dimension of E (∞) . In the second part we introduce within a general theory, a probabolistic justification for a compactification which reduces an infinite dimensional spacetime E (∞) (n=∞) to a four dimensional one (DT=n=4) . The effective Hausdorff dimension of this space is given by fn5 dimH E (∞) 〉=dH =4+φ3 where φ3 =1/ [4+φ3 ] is a PV number and φ=( 5 −1)/2 is the Golden Mean. The derivation makes use of various results from knot theory, four manifolds, noncommutative geometry, quasi periodic tiling and Fredholm operators. In addition some relevant analogies between E (∞) , statistical mechanics and Jones polynomials are drawn. This allows a better insight into the nature of the proposed compactification, the associated E (∞) space and the Pisot-Vijayvaraghavan number 1/φ3=4.236067977 representing its dimension. This dimension is in turn shown to be capable of a natural interpretation in terms of Jones knot invariant and the signature of four manifolds. This brings the work near to the context of Witten and Donaldson topological quantum field theory.

The Cabling conjecture states that Dehn surgery on a nontrivial knot K in S3 yields a reducible manifold only when K is a nontrivial cabled knot. One idea is to attack this problem with the techniques used by Gordon and Luecke in the Knot Complement Problem. This involves a combinatorial analysis of two intersecting planar graphs. In the context of reducible surgery, one of the two planar graphs will necessarily contain a Scharlemann cycle. So, we define a strict x-cycle to be any x-cycle which is not a Scharlemann cycle; likewise, for strict great x-cycles. We show that if the reducing sphere meets the core of the Dehn filling minimally, then strict great x-cycles are not permitted. Thus, strict great x-cycles can play a role similar to that of the Scharlemann cycle in the Knot Complement Problem. The obstruction of finding strict great x-cycles is considered an essential step in the program. A second conjecture states that Dehn surgery on a nontrivial knot K in S3 yields a manifold containing an embedded projective plane only when K is a nontrivial cabled knot. We show how the Gordon and Luecke technique can be applied towards this conjecture by considering the spherical boundary of a regular neighborhood of the projective plane. And if the projective plane is chosen to meet the core of the Dehn filling minimally, we show that strict great x-cycles are not permitted.

Let K be an unknotting number one knot. By calculating Casson's invariant for the 2-fold branched covering of S3 branched over K, we give some relations among the Jones polynomial, the signature, and the Conway polynomial of K, and prove that some knots are of unknotting number two.

This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We relate Fox 3-colorings to Jones and Kauffman polynomials of links and we use this connection to sketch the method of approximating the unknotting number of a knot. We discuss some elementary open problems in knot theory and show that Fox colorings can be useful in trying to solve them. Finally we demonstrate how analysis of Fox colorings can lead us to understand homology and the fundamental group of branched coverings along links., 28 pages, 27 figures

In this paper we show how the formula for the sl(2, C ), quantum invariant of the complement of a regular neighborhood of a link, can be proved in the context of topological quantum field theory with corners. As an application we will describe a short proof for the formulas for the colored Jones polynomials of torus knots.

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.

Based on the quantum invariant theory, the quantum phases, including the total phase as well as its dynamical and geometric parts, of Pancharatnam type are derived for a general spin in a time-dependent magnetic field, without the constraint of adiabatic, cyclic or unitary condition. The geometric meaning of geometric phase is expounded.

We extend the universal quantum invariant defined in [15] to an invariant of 3-manifolds with boundaries, and show that the invariant satisfies modified axioms of TQFT.

Mathematics Subject Classification (1991): 57M25, 57M05, 58G03, 57R571 IntroductionThe purpose of this paper is to establish the existence of irreducible SU(2) repre-sentations of knot complement fundamental groups near abelian representationswhere the equivariant knot signature changes. For knots in S

The quantizations of n identical or distinguishable particles, which are localized on closed orientable 3-manifolds, viewed as a three-fold branched covering of with a matching knot chosen as branching set, are classified up to unitary equivalence. The result connects the set of non-equivalent quantizations to knot theory.

The fact that a Yang-Baxter operator defines tensor representations of the Artin braid group has been used to construct knot invariants. The main purpose of this note is to extend the tensor representations of the Artin braid group to representations of the braid groupZ Bk associated to the Coxeter graphBk. This extension is based on some fundamental identities for the standardR-matrices of quantum Lie theory, here called four braid relations. As an application, tensor representations of knot algebras of typeB (Hecke, Temperley-Lieb, Birman-Wenzl-Murakami) are derived.

The linear skein theory for the Kauffman bracket was introduced by Lickorish [11,12]. It gives an elementary construction of quantum SU(2) invariant of 3-manifolds. In this paper we prove basic properties of the linear skein theory for quantum SU(3) invariant. By using them we give an elementary construction of quantum SU(3) invariant of 3-manifolds and prove topological invariance of the invariant along the construction.

We characterise the cabling operations on the weight systems of finite type knot invariants. The eigenvectors and eigenvalues of this family of operations are described over the cable eigenbasis. The action of immanent weight systems on general Feynman diagrams is considered, and the highest eigenvalue cabling eigenvectors are shown to be dual to the immanent weight systems. Using these results, we prove a recent conjecture of Bar-Natan and Garoufalidis on cablings of weight systems., Comment: LaTeX2e, 31 pages. This version to appear in the Journal of Knot Theory and it's Ramifications

We give an explicit algorithm for computing all of Vassiliev's knot invariants of order ⩽ n, which has been implemented on a computer. In giving a justification of the algorithm, we obtain a simple proof of the sufficiency of the topological 1-term and 4-term relations. We use an example of Taniyama to show that two singular knot diagrams with the same configuration cannot always be made isotopic by crossing changes alone.

We derive from the perturbation expansion of the Wilson loop expectation value in the Chern-Simons theory an explicit combinatorial expression for a third-order finite link invariant, thereby generalising the knot invariant considered in a previous article.

In this paper I present the Vassiliev invariant of degree 2 of a knot as a polynomial of degree 4 in gleams of a shadow presenting the knot. The coefficients of this polynomial involve Strangeness (a numerical characteristic of a generic immersion of the circle into the plane introduced by Arnold [1]) and a spherical index. The latter is a characteristic of a generic immersed circle on the sphere with three holes, which is invariant with respect to homotopy. I define it by an explicit combinatorial formula.

Let (V, Z) be a Topological Quantum Field Theory over a field f defined on a cobordism category whose morphisms are oriented n + 1-manifolds perhaps with extra structure (for example a p1 structure and banded link). Let (M, χ) be a closed oriented n + 1-manifold M with this extra structure together with χ ϵ H1(M). Suppose χ: H1(M) → Z is an epimorphism. Let M∞ denote the infinite cyclic cover of M given by χ. Consider a fundamental domain E for the action of the integers on M∞ bounded by lifts of a surface Σ dual to χ, and in general position. E can be viewed as a cobordism from Σ to itself. We give Turaev and Viro's proof of their theorem that the similarity class of the nonnilpotent part of Z(E) is an invariant. We give a method to calculate this invariant for the (Vp, Zp) theories of Blanchet, Habegger, Masbaum and Vogel when M is 0-framed surgery to S3 along a knot K. We give a formula for this invariant when K is a twisted double of another knot. We obtain formulas for the quantum invariants of branched covers of knots, and unbranched covers of 0-surgery to S3 along knots. We study periodicity among the quantum invariants of Brieskorn manifolds. We give an upper bound on the quantum invariants of branched covers of fibered knots. We also define finer invariants for pairs (M, χ) for TQFTs over Dedekind domains. We use these ideas to study isotopy invariants of banded links in S1 × S2.

V. Vassiliev [l] introduced a natural filtration in the space of finite order knot invariants. The corresponding grade space has a purely combinatorial description [2] as a space of functions u on the set of chord diagrams satisfying certain linear equations (one- and four-term relations).

The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number N. By the analysis of particularexamples, it is argued that, for a hyperbolic knot (link), the absolute valueof this invariant grows exponentially at large N, the hyperbolic volume of the knot (link) complement being the growth rate.

The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the converse; if the bounds are sharp then the diagram is semi-adequate. As a result, we use colored Jones link polynomials to extract an invariant that detects semi-adequate links and discuss some applications., To appear in Acta Math. Vietnamica (Proceedings of Hyperbolic Geometry and Quantum Topology in Nha Trang)

We define a "reduced" version of the knot Floer complex $CFK^-(K)$, and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer $d$-invariants of manifolds arising as surgeries on the knot $K$. As an application to connected sums, we prove that if a knot in the three-sphere admits an $L$-space surgery, it must be a prime knot. As an application of the computation of $d$-invariants, we show that the Alexander polynomial is a concordance invariant within the class of $L$-space knots, and show the four-genus bound given by the $d$-invariant of +1-surgery is independent of the genus bounds given by the Ozsv\'ath-Szab\'o $\tau$ invariant, the knot signature and the Rasmussen $s$ invariant., Comment: 41 pages, 14 figures; changed formatting, updated references, added some clarifying remarks, results unchanged

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is $\mathfrak{sl}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants., 99 pages. This is a significantly rewritten version of arXiv:1001.2020 and arXiv:1005.4559; both the exposition and proofs have been significantly improved. These earlier papers have been left up mainly in the interest of preserving references. v3: final version, to appear in Memoirs of the AMS. Proof of nondegeneracy moved to separate erratum

A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a uniform method to construct invariants for handlebody-links. We introduce the category $\mathcal{T}$ of handlebody-tangles and present it by generators and relations. The result tells us that every functor on $\mathcal{T}$ that gives rise to invariants is derived from what we call a quantum-commutative quantum-symmetric algebra in the target category. The example of such algebras of our main concern is finite-dimensional unimodular Hopf algebras. We investigate how those Hopf algebras give rise to handlebody-knot invariants., 25 Pages

The knot nomenclature in common use, summarized in Rolfsen's knot table [Rolfsen (1990) Knots and Links, American Mathematical Society], was not originally designed to distinguish between mirror images. This ambiguity is particularly inconvenient when studying knotted biopolymers such as DNA and proteins, since their chirality is often significant. In the present article, we propose a biologically meaningful knot table where a representative of a chiral pair is chosen on the basis of its mean writhe. There is numerical evidence that the sign of the mean writhe is invariant for each knot in a chiral pair. We review numerical evidence where, for each knot type K, the mean writhe is taken over a large ensemble of randomly chosen realizations of K. It has also been proposed that a chiral pair can be distinguished by assessing the writhe of a minimal or ideal conformation of the knot. In all cases examined to date, the two methods produce the same results.

Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $(m,n)$-mosaic is an $m \times n$ matrix of mosaic tiles which are $T_0$ through $T_{10}$ depicted as below, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot $(m,n)$-mosaics are there. $D_{m,n}$ denotes the total number of all knot $(m,n)$-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of $D_{m,n}$ for $4 \leq m \leq n \leq 6$ as below. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics. \begin{center} \begin{tabular}{|c|r|r|r|} \hline $D_{m,n}$ & $n=4$ & $n=5$ & $n=6$ \\ \hline $m=4$ & $2594$ & $54,226$ & $1,144,526$ \\ \hline $m=5$ & & $4,183,954$ & $331,745,962$ \\ \hline $m=6$ & & & $101,393,411,126$ \\ \hline \end{tabular} \end{center}, 13 pages, 9 figures, 1 table

In these notes we collect some results about finite dimensional representations of $U_q(\mathfrak{gl}(1|1))$ and related invariants of framed tangles which are well-known to experts but difficult to find in the literature. In particular, we give an explicit description of the ribbon structure on the category of finite dimensional $U_q(\mathfrak{gl}(1|1))$-representation and we use it to construct the corresponding quantum invariant of framed tangles. We explain in detail why this invariant vanishes on closed links and how one can modify the construction to get a nonzero invariant of framed closed links. Finally we show how to obtain the Alexander polynomial by considering the vector representation of $U_q(\mathfrak{gl}(1|1))$., Comment: This is a corrected and revised version, to be published on Arkiv f\"or Matematik. This is part of the author's PhD Thesis

In this paper we establish a relation between an appropriate version of the Casson-Gordon invariant of a satellite knot and those of its orbit and companion. We note that in some cases the contribution from, the companion falls. This gives a way to construct algebraically but not smoothly slice knots.