Knot Floer Homology of (1, 1)-Knots

We present a combinatorial method for a calculation of the knot Floer homology of (1, l)-knots, and then demonstrate it for nonalternating (1, 1)-knots with 10 crossings and the pretzel knots of type (−2,m, n). Our calculations determine the unknotting numbers and 4-genera of the pretzel knots of this type.Mathematics Subject Classiffications (2000). 57M27, 57M25     

The Canonical Genus of a Classical and Virtual Knot

A diagram D of a knot defines the corresponding Gauss Diagram G _D . However, not all Gauss diagrams correspond to the ordinary knot diagrams. From a Gauss diagram G we construct closed surfaces F _G and S _G in two different ways, and we show that if the Gauss diagram corresponds to an ordinary knot diagram D, then their genus is the genus of the canonical Seifert surface associated to D. Using these constructions we introduce the virtual canonical genus invariant of a virtual knot and find estimates on the number of alternating knots of given genus and given crossing number.     

A New Way to Represent Links. One-Dimensional Formalism and Untangling Technology

An alternative link representation different from planar diagrams is discussed. Isotopy classes of unordered nonoriented links are realized as central elements of a monoid presented explicitly by a finite number of generators and relations. The group presented by two generators and three relations [[a,b],a ² ba ^–2]=[[a,b],b ² ab ^–2]=[[a,b],[a ^–1,b ^–1]]=1, where [x,y]=xyx ^–1 y ^–1, is proved to have a commutator subgroup isomorphic to the braid group on infinitely many strands. A new partial algorithm for unknot recognition is constructed. Experiments show that the algorithm allows the untangling of unknots whose planar diagram has hundreds of crossings. Here 'untangling' means 'finding an isotopy to the circle'.     

q-Series and L-functions related to half-derivatives of the Andrews-Gordon identity

Studied is a generalization of Zagier’s q-series identity. We introduce a generating function of L-functions at non-positive integers, which is regarded as a half-differential of the Andrews-Gordon q-series. When q is a root of unity, the generating function coincides with the quantum invariant for the torus knot. 2000 Mathematics Subject Classification Primary—11F67, 57M27, 05A30, 11F23     

The Volume of Hyperbolic Alternating Link Complements

If a hyperbolic link has a prime alternating diagram D, thenwe show that the link complement's volume can be estimated directlyfrom D. We define a very elementary invariant of the diagramD, its twist number t(D), and show that the volume lies betweenv₃(t(D) – 2)/2 and v₃(10t(D) – 10), where v₃ isthe volume of a regular hyperbolic ideal 3-simplex. As a consequence,the set of all hyperbolic alternating and augmented alternatinglink complements is a closed subset of the space of all completefinite-volume hyperbolic 3-manifolds, in the geometric topology.2000 Mathematics Subject Classification 57M25, 57N10.     

Torus Knots and Dunwoody Manifolds

We obtain an explicit representation as Dunwoody manifolds of all cyclic branched coverings of torus knots of type (p,mp±1), with p > 1 and m > 0.     

Upper bound for the alternation number of a torus knot

We give an upper bound for the alternation number of a torus knot which is of either 3-, 4-, or 5-braid or of other special types. Using the inequality relating the alternation number, signature, and Rasmussen s-invariant, discovered by Abe, we determine the alternation numbers of the torus knots T(3,l), , and T(4,5). Also, for any positive integer k we construct infinitely many 3-braid knots with alternation number k.     

Legendrian Knots in Overtwisted Contact Structures on S3

We study the problem of classifying Legendrian knots in overtwisted contact structures on S ³. The question is whether topologically isotopic Legendrian knots have to be Legendrian isotopic if they have equal values of the well-known invariants rot and tb. We give positive answer in the case that there is an overtwisted disc intersecting none of the knots and we construct an example of a knot intersecting each overtwisted disc (this provides a counterexample to the conjecture of Eliashberg). Our proof needs some results on the structure of the group of contactomorphisms of S ³. We divide the subgroup Cont₊(S ³, ) of coorientation-preserving contactomorphisms for an overtwisted contact distribution into two classes.     

Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer . In the present paper, we study the following more general situation. Given two integers m and n, how are knots K ₁ and K ₂ related such that the m-fold cyclic branched covering of K ₁ coincides with the n-fold cyclic branched covering of K ₂. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S ³. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds). Received: December 7, 1999; revised version: May 22, 2000     

On Invariants of Virtual Links

We propose a new method of generalizing classical link invariants for the case of virtual links. In particular, we have generalized the knot quandle, the knot fundamental group, the Alexander module, and the coloring invariants. The virtual Alexander module leads to a definition of VA-polynomial that has no analogue in the classical case (i.e. vanishes on classical links).     

Epstein–Penner Decompositions and Thrice Punctured Spheres in Hyperbolic 3-Manifolds

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.     

Toroidal Dehn surgery on hyperbolic knots and hitting number

In this paper, we prove that for any positive even integer m, there exists a hyperbolic knot such that its longitudinal Dehn surgery yields a 3-manifold containing a unique separating, incompressible torus, which meets the core of the attached solid torus in m points minimally.     

The Second Bounded Cohomology of a Group Acting on a Gromov-Hyperbolic Space

Suppose a group G acts on a Gromov-hyperbolic space X properlydiscontinuously. If the limit set L(G) of the action has atleast three points, then the second bounded cohomology groupof is infinite dimensional. For example, if M is a complete, pinched negatively curved Riemannianmanifold with finite volume, then is infinite dimensional. As an application, we show that ifG is a knot group with GZ, then is infinite dimensional. 1991 Mathematics Subject Classification:primary 20F32; secondary 53C20, 57M25.     

Inhomogeneous Current States in a Gauged Two-Component Ginzburg-Landau Model

We consider the energy bounds of inhomogeneous current states in doped antiferromagnetic insulators in the framework of the two-component Ginzburg-Landau model. Using the formulation of this model in terms of the gauge-invariant order parameters (the unit vector n, spin stiffness field ρ², and particle momentum c), we show that this strongly correlated electron system involves a geometric small parameter that determines the degree of packing in the knots of filament manifolds of the order parameter distributions for the spin and charge degrees of freedom. We find that as the doping degree decreases, the filament density increases, resulting in a transition to an inhomogeneous current state with a free energy gain.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 182–189, July, 2005.     

The writhe of permutations and random framed knots

We introduce and study the writhe of a permutation, a circular variant of the well‐known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non‐Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model, studied with Hass, Linial, and Nowik (Discrete Comput Geom, 2016). The distribution of the framing in this model is equivalent to the writhe of random permutations. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 121–142, 2017     

Application of computer algebra to Jones polynomials

In this paper we apply computer algebra (Maple) techniques to calculate Jones polynomial of graphs of K(2,q)-Torus knots. For this purpose, a computer program was developed. When a positive integer q is given, the program calculate Jones polynomial of graph of K(2,q)-Torus knots.     

An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsváth–Szabó contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston–Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than S³ with negative maximal Thurston–Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsváth–Szabó invariants, then any fibered knot can realize the classical Eliashberg–Bennequin bound in at most one of these contact structures.     

The Neumann-Siebenmann invariant and Seifert surgery

We discuss some relations between the invariant originated in Fukumoto-Furuta and the Neumann-Siebenmann invariant for the Seifert rational homology 3-spheres. We give certain constraints on Seifert 3-manifolds to be obtained by surgery on knots in homology 3-spheres in terms of these invariants.Mathematics Subject Classification (2000): 57M27, 57N13, 57N10Dedicated to Professor Yukio Matsumoto for his 60^th birthday     

On the distribution of comparisons in sorting algorithms

We considered the following natural conjecture: For every sorting algorithm every key will be involved in(logn) comparisons for some input. We show that this is true for most of the keys and prove matching upper and lower bounds. Every sorting algorithm for some input will involven–n /2+1 keys in at leastlog₂ n comparisons,>0. Further, there exists a sorting algorithm that will for every input involve at mostn–n ^/c keys in greater thanlog₂ n comparisons, wherec is a constant and>0. The conjecture is shown to hold for natural algorithms from the literature.