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We discuss the development of the Kakimizu complex as well as the mathematical techniquesthat have been developed to study it. By using these techniques we determine the Kakimizucomplex for a new class of links which are the result of plumbing an n-times full twisted band,n ≥ 2 to a Seifert surface for a special alternating link. We also show how the maximalsimplices of the Kakimizu complex for the resulting link can be directly obtained from themaximal simplices of the original special alternating link.

Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of $n$-component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates and thus generalizing the known linking number bound. In the process, we collect several facts about finite type invariants and ropelength/crossing number of knots. We give examples of families of knots, where such estimates behave better than the well-known knot-genus estimate., 16 pages, 6 figures (accepted for publication), corrected: proofs of Theorem B, Lemma F, and constants in the estimates of Theorem A

International audience; The present volume consists of a collection of essays dedicated to Vladimir Turaev. The essays cover the large spectrum of topics in which Turaev has been interested, including knot and link invariants, quantum representations, TQFTs, state sum constructions, geometric structures on knot complements, Kleinian groups, geometric group theory and its relationship with 3-manifolds, mapping class groups, operads, mathematical physics, Grothendieck’s program, the philosophy of mathematics, and several other topics. At the same time, this volume will give an overview of topics that are at the forefront of current research in topology and geometry. Some of the essays are research articles and contain new results, sometimes answering questions that were raised by Turaev. The rest of the essays are surveys that will introduce the reader to some key ideas in the field. The authors of the essays are: Athanase Papadopoulos, Jean-Claude Hausmann, Charles Livingston, Norbert A’Campo, Julia Viro and Oleg Viro, Louis H. Kauffman, Julien Marché, Rinat Kashaev, Jun Murakami, Anna Beliakova and Kazuhiro Hikami, Christian Blanchet and Marco De Renzi, Louis-Hadrien Robert and Emmanuel Wagner, Sergey M. Natanzon, Stavros Garoufalidis and Rinat Kashaev, Louis Funar, Toshitake Kohno, Nariya Kawazumi, Yusuke Kuno, Gwénaël Massuyeau and Shunsuke Tsuji, Valentin Poénaru, Nikolay Abrosimov and Alexander Mednykh, Charalampos Charitos, Ken’ichi Ohshika, Gourab Bhattacharya and Maxim Kontsevich, Noémie C. Combe, Yuri I. Manin and Matilde Marcolli, A. Muhammed Uludag, Sumio Yamada, Vassiliki Farmaki and Stelios Negrepontis, Arkady Plotnitsky.

Let $\Sigma$ be a compact oriented surface. The Dehn twist along every simple closed curve $\gamma \subset \Sigma$ induces an automorphism of the fundamental group $\pi$ of $\Sigma$. There are two possible ways to generalize such automorphisms if the curve $\gamma$ is allowed to have self-intersections. One way is to consider the `generalized Dehn twist' along $\gamma$: an automorphism of the Malcev completion of $\pi$ whose definition involves intersection operations and only depends on the homotopy class $[\gamma]\in \pi$ of $\gamma$. Another way is to choose in the usual cylinder $U:=\Sigma \times [-1,+1]$ a knot $L$ projecting onto $\gamma$, to perform a surgery along $L$ so as to get a homology cylinder $U_L$, and let $U_L$ act on every nilpotent quotient $\pi/\Gamma_{j} \pi$ of $\pi$ (where $\Gamma_j\pi$ denotes the subgroup of $\pi$ generated by commutators of length $j$). In this paper, assuming that $[\gamma]$ is in $\Gamma_k \pi$ for some $k\geq 2$, we prove that (whatever the choice of $L$ is) the automorphism of $\pi/\Gamma_{2k+1} \pi$ induced by $U_L$ agrees with the generalized Dehn twist along $\gamma$ and we explicitly compute this automorphism in terms of $[\gamma]$ modulo ${\Gamma_{k+2}}\pi$. As applications, we obtain new formulas for certain evaluations of the Johnson homomorphisms showing, in particular, how to realize any element of their targets by some explicit homology cylinders and/or generalized Dehn twists., Comment: 45 pages; minor modifications with respect to the first version

The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod 4, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, torus knots, and Montesinos knots, as well as for several families of two-components links., Comment: 59 pages. Corrected a grading error in Lemma 2.5, which affected calculations for some of the knots

This paper is devoted to the study of the spectral properties of Dirac operators on the three-sphere with singular magnetic fields supported on smooth, oriented links. As for Aharonov–Bohm solenoids in the Euclidean three-space, the flux carried by an oriented knot features a 2 π-periodicity of the associated operator. For a given link, one thus obtains a family of Dirac operators indexed by a torus of fluxes. We study the spectral flow of paths of such operators corresponding to loops in this torus. The spectral flow is in general nontrivial. In the special case of a link of unknots, we derive an explicit formula for the spectral flow of any loop on the torus of fluxes. It is given in terms of the linking numbers of the knots and their writhes.

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $$\vec {F}$$ and $$\vec {T}$$ of the Thompson groups F and T, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones–Thompson subgroups. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups $$\vec {F}$$ and $$\vec {T}$$ . Within this framework, we discuss an alternative approach that relies on some partition function interpretation of the Jones polynomial, and also exhibit more examples associated with other link invariants, notably the two-variable Kauffman polynomial and the HOMFLY polynomial. In the unoriented case, extending our previous results, we also show by similar methods that certain evaluations of the Tutte polynomial and of the Kauffman bracket, suitably renormalized, yield functions of positive type on T.

We construct a Hennings-type logarithmic invariant for restricted quantum $sl(2)$ at a$2p^{th}$ root of unity. This quantum group $U$ is not quasitriangular and hence not ribbon, but factorizable. The invariant is defined for a pair: a $3–$manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into two parts colored by central elements and by trace classes, or elements in the $0^{th}$ Hochschild homology of $U$, respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of U, and the modified trace introduced by the third author with his collaborators and computed on tensor powers of the regular representation. Our invariant is a colored extension of the logarithmic invariant constructed by Jun Murakami.

International audience; Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson–Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine– Tristram signatures to compute these bounds and produce explicit examples.

We study the class L of link types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimal number of crossings or edges in a projection of L ∈ L . Further, we enumerate (both exactly and asymptotically) all connected K4-minor-free link diagrams, all minimal connected K4-minor-free link diagrams, and all K4-minor-free diagrams of the unknot.

We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the problem by Kawamuro., Comment: 10 pages, 13 figures; expanded background, added figures

We propose a new heuristic for nonlinear global optimization combining a variable neighborhood search framework with a modified trust-region algorithm as local search. The proposed method presents the capability to prematurely interrupt the local search if the iterates are converging to a local minimum that has already been visited or if they are reaching an area where no significant improvement can be expected. The neighborhoods, as well as the neighbors selection procedure, are exploiting the curvature of the objective function. Numerical tests are performed on a set of unconstrained nonlinear problems from the literature. Results illustrate that the new method significantly outperforms existing heuristics from the literature in terms of success rate, CPU time, and number of function evaluations.

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach., Comment: 19 pages, 9 figures, minor changes

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials., Comment: 19 pages

We give an explicit formula for the fact given by Links and Gould that a one variable reduction of the LG polynomial coincides with a one variable reduction of the Kauffman polynomial. This implies that the crossing number of an adequate link may be obtained from the LG polynomial by using a result of Thistlethwaite. We also give some evaluations of the LG polynomial.

For a link K, let L ( K ) denote the ropelength of K and let Cr ( K ) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L ( K ) in terms of Cr ( K ) . It is well known that there exist positive constants c 1 , c 2 such that for any link K, c 1 ⋅ ( Cr ( K ) ) 3 / 4 ⩽ L ( K ) ⩽ c 2 ⋅ ( Cr ( K ) ) 3 / 2 . In this paper, we show that any closed braid with n crossings can be realized by a unit thickness rope of length at most of the order O ( n 6 / 5 ) . Thus, if a link K admits a closed braid representation in which the number of crossings is bounded by a ( Cr ( K ) ) for some constant a ⩾ 1 , then we have L ( K ) ⩽ c ⋅ ( Cr ( K ) ) 6 / 5 for some constant c > 0 which only depends on a. In particular, this holds for any link that admits a reduced alternating closed braid representation, or any link K that admits a regular projection in which there are at most O ( Cr ( K ) ) crossings and O ( Cr ( K ) ) Seifert circles.

For DNA molecules, topological complexity occurs exclusively as the result of knotting or linking of the polynucleotide backbone. By contrast, while a few knots and links have been found within the polypeptide backbones of some protein structures, non-planarity can also result from the connectivity between a polypeptide chain and inter- and intra-chain linking via cofactors and disulfide bonds. In this article, we survey the known types of knots, links, and non-planar graphs in protein structures with and without including such bonds and cofactors. Then we present new examples of protein structures containing Möbius ladders and other non-planar graphs as a result of these cofactors. Finally, we propose hypothetical structures illustrating specific disulfide connectivities that would result in the key ring link, the Whitehead link and the 51 knot, the latter two of which have thus far not been identified within protein structures.

A new state-sum formula for the evaluation of the Yang-Mills measure in the Kauffman bracket skein algebra of a closed surface is derived. The formula extends the Kauffman bracket to diagrams that lie in surfaces other than the plane. It also extends Turaev's shadow world invariant of links in a circle bundle over a surface away from roots of unity. The limiting behavior of the Yang-Mills measure when the complex parameter approaches -1 is studied. The formula is applied to compute integrals of simple closed curves over the character variety of the surface against Goldman's symplectic measure., Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-17.abs.html

Our research concerns how knots behave under crossing changes. In particular, we investigate a partial ordering of alternating knots that results from performing crossing changes. A similar ordering was originally introduced by Kouki Taniyama in the paper “A partial order of knots”. We amend Taniyama’s partial ordering and present theorems about the structure of our ordering for more complicated knots. Our approach is largely graph theoretic, as we translate each knot diagram into one of two planar graphs by checkerboard coloring the plane. Of particular interest are the class of knots known as pretzel knots, as well as knots that have only one direct minor in the partial ordering.

In a recent paper Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of $F$. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of $N$-colourings and the Tutte polynomial, can be viewed as positive definite functions on $F$., Comment: To appear in Journal of Knot Theory and Its Ramifications

To show that Ramsey theorem for spatial graphs without local knots does not hold in general, we construct a spatial embedding of Kn,n which has no local knots on edges and which contains any subdivision of a given nonsplittable 2-component link.

The well-known technique of n -coloring a diagram of an oriented link l is generalized using elements of the circle T for colors. For any positive integer r , the more general notion of a ( T , r )-coloring is defined by labeling the arcs of a diagram D with elements of the torus T r−1 . The set of ( T , r )-colorings of D is an abelian group, and its quotient by the connected component of the identity is isomorphic to the torsion subgroup of H 1 (M r (l) ; Z). Here M r (l) denotes the r -fold cyclic cover of S 3 branched over the link l . Results about braid entropy are obtained using techniques of symbolic dynamical systems.

We consider smooth links in S3 which consist of a countably infinite collection of tamely embedded circles having the property of being locally straight. We show that in general there are no restrictions on the types of accumulations that may appear in terms of the knot types: e.g., trefoils may accumulate on figure-eights in a smooth manner. Our techniques rely on properties of “universal templates”, a class of branched 2-manifolds. Our results apply to closed orbits of flows on S3.

In this note, we describe a procedure to construct generalized complex structures with an arbitrarily large number of type change loci on products of the circle with a connected sum of closed 3-manifolds. The loci need not be isotopic., Comment: Revised version: improved exposition, added a reference and a result

We define the notion of a braided link cobordism in $S^3 \times [0,1]$, which generalizes Viro's closed surface braids in $\mathbb{R}^4$. We prove that any properly embedded oriented surface $W \subset S^3 \times [0,1]$ is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary when $\partial W$ already consists of closed braids. These surfaces are closely related to another notion of surface braiding in $D^2 \times D^2$, called braided surfaces with caps, which are a generalization of Rudolph's braided surfaces. We mention several applications of braided surfaces with caps, including using them to apply algebraic techniques from braid groups to studying surfaces in 4-space, as well as constructing singular fibrations on smooth 4-manifolds from a given handle decomposition., Comment: 18 pages, 13 figures. The previous version of this paper contained sections outlining the construction of broken Lefschetz fibrations via braided surface techniques. These sections will now appear in a separate paper, along with additional examples

International audience; The book contains surveys on geometry and topology, more specifically on hyperbolic geometry, three-manifold topology, representation theory of fundamental groups of surfaces and of three-manifolds, dynamics on the hyperbolic plane with applications to number theory, Riemann surfaces, Teichmüller theory, Lie groups and asymptotic geometry. The surveys are accessible to graduate students and they will be useful to researchers. They can also be used as course lecture notes. The titles and authors are: (1) Notes on non-Euclidean geometry, by Norbert A'Campo and Athanase Papadopoulos, pp. 1-182 ; (2) Crossroads between hyperbolic geometry and number theory, by Françoise Dal'Bo, pp. 183-232 ; (3) Introduction to origamis in Teichmüller space, by Frank Herrlich, pp. 233-253 ; (4) Five lectures on 3-manifold topology, by Philipp Korablev and Sergey V. Matveev, pp. 255-284 ; (5) An introduction to globally symmetric spaces, by Gabriele Link, pp. 285-332 ; (6) Geometry of the representation spaces in SU(2), by Julien Marché, pp. 333-370 ; (7) Algorithmic construction and recognition of hyperbolic 3-manifolds, links, and graphs, by Carlo Petronio, pp. 371-404 ; (8) An introduction to asymptotic geometry, by Viktor Schroeder, pp. 405-454.

We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere., Comment: 10 pages, v3: The organization was changed. This paper focuses on proving the formula of the twisted Alexander polynomial for finite abelian covering spaces, typos corrected and the main statement and proof were improved, to appear in Algebraic & Geometric Topology

This paper addresses two problems in the skein theory of homotopy spheres first posed by P. Traczyk. Solutions to both problems are obtained for a large class of manifolds and, since one of the basic techniques used requires the first homology group of the ambient manifold to be torsion free, the extent to which this hypothesis is actually necessary is further explored.

Inversions are various structural possibilities of a kinematic chain. The number of inversions depends on the number of links of a kinematic chain. At the stage of structural synthesis, identification of distinct structural inversions of a particular type of kinematic chain is necessary. Various researchers have proposed methods for identification of distinct inversions. Present method based on Link joint connectivity is proposed to identify the distinct inversions of a planar kinematic chain. Method is tested successfully on single degree and multiple degree of freedom planar kinematic chains. ABSTRAK: Penyonsangan merupakan kebarangkalian pelbagai struktur suatu rangkaian kinematik. Jumlah songsangan bergantung kepada jumlah hubungan suatu rangkaian kinematik. Pada peringkat sintesis struktur, pengenalan songsangan struktur yang berbeza untuk suatu jenis rangkaian kinematik adalah perlu. Ramai penyelidik telah mencadangkan pelbagai kaedah pengenalan songsangan yang berbeza. Kaedah terkini berdasarkan hubungan kesambungan bersama telah dicadangkan untuk mengenalpasti songsangan yang berbeza dalam suatu satah rangkaian kinematik.

We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

We consider for 2-complexes K in 4-manifolds N, the problem of effecting by a geometric deformation of K in N, the free reduction of the relator words in the presentation P K associated with K. In order to get a more clear picture, we restrict our attention to deformations of K in N that hold the 1-skeleton K1 of K fixed. For a large class of 2-complexes, called split 2-complexes, a class that includes all 2-complexes K contained in bicollared 3-manifold sections of the 4-manifolds N, we show that the problem is an obstruction problem, with obstructions measured by finite families of links in a handlebody. The sought for deformation effecting free reduction can be found if and only if one of the links in the family is trivial. We also provide a fairly complete picture on what links can arise as obstruction links.

We show that the Kakimizu complex of a knot may be locally infinite, answering a question of Przytycki--Schultens. We then prove that if a link $L$ only has connected Seifert surfaces and has a locally infinite Kakimizu complex then $L$ is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0., Comment: 9 pages, 5 figures; v2 minor has minor changes incorporating referee's comments. To appear in Algebraic & Geometric Topology

In this paper we define a partial ordering of knots and links using a special property derived from their minimal diagrams. A link [Formula: see text] is called a predecessor of a link [Formula: see text] if [Formula: see text] and a diagram of [Formula: see text] can be obtained from a minimal diagram D of [Formula: see text] by a single crossing change. In such a case, we say that [Formula: see text]. We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.