Space Of Knots Research Articles

The Jones slopes of a knot

The paper introduces the slope conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, the knots with at most 9 crossings, the torus knots and the (−2,3,n) pretzel knots.

The Length Scale of $3$-Space Knots, Ephemeral Knots, and Slipknots in Random Walks

The Length Scale of $3$-Space Knots, Ephemeral Knots, and Slipknots in Random Walks

A Technique of Bilateral Inguinal Hernia Repair Using 10 mm Single Port Access and Bioresorbable Composite Mesh Fixed with Endoclose Sutures: Three Cases Reported

Purpose: To report a novel technique of laparoscopic 10 mm Single Port Access IntraPeritoneal Onlay Mesh (SPA-IPOM) using bioresorbable composite mesh fixed with Endoclose suture (percutaneous subcutaneous suture) in 3 cases of bilateral inguinal hernia. Methods: Laparoscopic SPA-IPOM is done through a 10 mm trocar with one 10 mm instrument that has 5 mm optical lens and 5 mm channel for grasper. After inserting 10 mm trocar at umbilicus using semi-open technique, intraperitoneal anatomical landmark of inguinal her-nia is identified. A 10 × 15 cm pre-tied bioresorbable composite mesh is then placed to cover hernia defect and all three potential area of indirect, direct and femoral hernia. Using Endoclose? needle, each pair of pre-tied sutures is retrieved percutaneously through a needle wound and extracorporeal tied with knot in subcutaneous space. After the upper half of mesh is sutured to the posterior surface of abdominal wall, the lower half of mesh is fixed by hernia tacker to Symphysis Pubis, Cooper Ligament and Iliopubic tract. Re-sults: Three men, average 48 year olds were operated by laparoscopic 10 mm SPA-IPOM for bilateral in-guinal hernia repair using bioresorbable composite mesh. Average operative time was 36 minutes. No imme-diate complication. All patients were discharge on the 2nd post-operative day and average 6 months follow up has no recurrence. Conclusions: Laparoscopic SPA-IPOM is an optional operation and is much easier to be performed. Benefits include operative time saving, cosmesis, early discharge and early return to work. Bio-resorbable composite mesh prevents bowel adhesion, however, is much more expensive. Long term follow up study for complications and recurrence is needed.

On Floer homology and the Berge conjecture on knots admitting lens space surgeries

We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge’s construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a threesphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.

Topology of knot spaces in dimension 3

This paper is a computation of the homotopy type of K, the space of long knots in R^3, the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we `enumerate' the components via the companionship trees associated to the knot. The knots with the simplest companionship trees are: the unknot, torus knots, and hyperbolic knots. The homotopy-type of these components of K were computed by Hatcher. In the case the companionship tree has height, we give a fibre-bundle description of those components of K, recursively, in terms of the homotopy types of `simpler' components of K, in the sense that they correspond to knots with shorter companionship trees. The primary case studied in this paper is the case of a knot which has a hyperbolic manifold contained in the JSJ-decomposition of its complement.

ARTIN'S BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE

The purpose of this article is to describe connections between the loop space of the 2-sphere and Artin's braid groups. The current article exploits Lie algebras associated with Vassiliev invariants in the work of Kohno (Linear representations of braid groups and classical Yang-Baxter equations, Cont. Math. 78 (1988), 339–369 and Vassiliev invariants and de Rham complex on the space of knots, Symplectic Geometry and Quantization, Contemp. Math. 179 (1994), Am. Math. Soc. Providence, RI, 123–138), and provides connections between these various topics. Two consequences are as follows: the homotopy groups of spheres are identified as ‘natural’ sub-quotients of free products of pure braid groups, and an axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected CW-complexes.

On the knot complement problem for non-hyperbolic knots

This paper explicitly provides two exhaustive and infinite families of pairs ( M , k ) , where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic to M, by a non-trivial Dehn surgery. Then, we observe the uniqueness of such knot in such lens space, the uniqueness of the slope, and that there is no preserving homeomorphism between the initial and the final M's. We obtain further that Seifert fibered knots, except for the axes, and satellite knots are determined by their complements in lens spaces. An easy application shows that non-hyperbolic knots are determined by their complement in atoroidal and irreducible Seifert fibered 3-manifolds.

The topology of spaces of knots: cosimplicial models

We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial. These models build on the calculus of isotopy functors and are weakly homotopy equivalent to knot spaces when the ambient dimension is greater than three. The mapping space model, and the evaluation map on which it builds, is suitable for analysis through differential topology. The cosimplicial model gives rise to spectral sequences which converge to ohomology and homotopy groups of spaces of knots when they are connected. We explicitly identify and establish vanishing lines in these spectral sequences.

A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber" classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated differential forms along the fiber to get knot invariants. By doing this "integration" homotopy-theoretically, we are able to produce integral cohomology classes. We then show how this integration is compatible with the homology operations on the space of long knots, as studied by Budney and Cohen. In particular we derive a product formula for evaluations of cohomology classes on homology classes, with respect to connect-sum of knots.

Orbit configuration spaces associated to discrete subgroups of [formula omitted

The purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” obtained from a group G acting freely, and properly discontinuously on the upper half-plane H 2 . The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to 1. the Lie algebra obtained from the higher homotopy groups of analogous constructions associated to H 2 × C q modulo torsion, as well as 2. the Lie algebra obtained from horizontal chord diagrams for surfaces. The resulting Lie algebras are similar to those studied in [T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math. 78 (1988) 339–363; T. Kohno, Vassiliev invariants and de Rham complex on the space of knots, Contemp. Math. 179 (1994) 123–138; T. Kohno, Elliptic KZ system, braid groups of the torus and Vassiliev invariants, Topology and its Applications 78 (1997) 79–94; D.C. Cohen, Monodromy of fiber-type arrangements and orbit configuration spaces, Forum Math. 13 (2001) 505–530; F.R. Cohen, M. Xicoténcatl, On orbit configuration spaces associated to the Gaussian integers: homology and homotopy groups, Topology Appl. 118 (2002) 17–29; E. Fadell and S. Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topol. Methods Nonlinear Anal. J. Julius Schauder Center 11 (1998), 249–271; E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces, in: Springer Monographs in Mathematics, Springer-Verlag, 2001; F.R. Cohen and T. Sato, On groups of morphisms of coalgebras, (submitted for publication)]. The structure of a related graded Poisson algebra defined below and obtained from an analogue of the infinitesimal braid relations parametrized by G is also addressed.

Invariants in Low-Dimensional Topology

The workshop Invariants in low-dimensional topology , organised by Louis H. Kauffman (Chicago), Simon A. King (Jena), Vassily O. Manturov (Moscow) and Jozef Przytycki (Washington) was held May 4th–May 10th, 2008. This meeting was attended by 46 participants, including various researchers with recent PhD, one PhD student and one graduate student. The main objective of this workshop was to look intensively at the present state of the art of invariants in low dimensional topolgy, particularly invariants of knots and three dimensional manifolds. This field is rapidly growing with a remarkable influx of new ideas and techniques. This activity was originally inspired in the 1980's by the discovery of the Jones polynomial. Vaughan Jones formulated his invariant in terms of braids and von Neumann algebras and also gave a formula for it via a skein relation (a relation among knot and link diagrams) that was a clear generalization of the Conway skein relation for the classical Alexander polynomial. This led at once to the discovery by many people of a generalization called the Homflypt polynomial, and then sometime later another skein theoretic generalization – the Kauffman polynomial – and to a general skein theory for knots in three-manifolds (skein modules and skein algebras) due to Jozef Przytycki. The Jones polynomial is related to statistical mechanics. A key notion from statistical mechanics that was imported here was the idea to write an invariant in analogy to a partition function. This means that the invariant is expressed as a sum over combinatorial states related either to a link diagram or a triangulation of a three-manifold. The first example of this is the state sum for the Jones polynomial called Kauffman bracket. The state sums generalize to sums involving solutions to the Yang-Baxter Equation (originally used in statistical mechanics) and this in turn led to constructions of invariants from quantum groups and Hopf algebras. Then in the latter part of the 1980's ideas from quantum field theory (and the work of Witten) crystallized in the notion of Topological Quantum Field Theories (due to Witten and Atiyah) and corresponding invariants of knots and three-manifolds. At about the same time Vassiliev and Goussarov introduced new ideas about the space of knots and associated invariants that were quickly reformulated through the work of Birman, Lin, Kontsevich and Bar-Natan yielding deep relationships with Lie algebras and new ways to understand the previous constructions involving quantum groups. In 1996 Kauffman introduced a generalization of knot theory called virtual knot theory, which is given diagrammatically, and is equivalent to a stabilized theory of knots in thickened surfaces. At the same time there was an evolution in the concept of the fundamental group of the knot complement with Joyce's introduction of the quandle in 1979 and Matveev's independent discovery of this structure (distributive groupoids). After 2000 Fenn, Kauffman, Manturov and others generalized the quandle to a stronger invariant called the biquandle which is very important for virtual knots. In 1998 Khovanov introduced a categorification of the Jones polynomial. This means that he constructed a homology theory based on link diagrams whose graded Euler characteristic equals the original Jones polynomial. This led eventually to an explosion of significant results and new constructions. Khovanov and Rozansky found ways to categorify the Homflypt polynomial, and at this conference Rozansky announced the categorification of the Kauffman 2-variable polynomial. After Khovanov's original discovery, a categorification of the Alexander polynonmial (Knot Floer Homology) was found by Ozsvath and Szabo (2002) and eventually made purely combinatorial by Manolescu, Ozsvath and Sarkar. This categorification of the Alexander polynomial has led one of the most startling results in knot theory: The minimal genus of an orientable spanning surface for a knot, can be calculated from the Knot Floer homology, and this is a finite combinatorial matter in terms of a diagram for the knot. This conference collected many active researchers in this field with talks on all of the topics mentioned above. We shall now mention the main themes of the workshop and the participants who were influential or gave talks on these themes: In our opinion, many of the results in the conference were new and significant. Having the conference at the Mathematisches Forschungsinstitute enabled the participants not only to listen to new results but also to the opportunity to converse and do mathematics together. The conference was dedicated to the 60th birthday of Oleg Yannovich Viro.

On the homology of the space of knots

Consider the space of long knots in ℝn, Kn,1. This is the space of knots as studied by V Vassiliev. Based on previous work [Budney: Topology 46 (2007) 1–27], [Cohen, Lada and May: Springer Lecture Notes 533 (1976)] it follows that the rational homology of K3,1 is free Gerstenhaber–Poisson algebra. A partial description of a basis is given here. In addition, the mod–p homology of this space is a free, restricted Gerstenhaber–Poisson algebra. Recursive application of this theorem allows us to deduce that there is p–torsion of all orders in the integral homology of K3,1. This leads to some natural questions about the homotopy type of the space of long knots in ℝn for n>3, as well as consequences for the space of smooth embeddings of S1 in S3 and embeddings of S1 in ℝ3.

NORMAL HOLONOMY AND WRITHING NUMBER OF SMOOTH KNOTS

The normal holonomy of a smooth knot in Euclidean space is a geometric invariant with value in the unit circle which is closely related to the writhing number. Here we prove that normal holonomy fibers the space of smooth knots, except for possible singularities on the set of round knots. We deduce results on the existence of isotopies of constant holonomy, writhe and twist.

Homotopy Approximations to the Space of Knots, Feynman Diagrams, and a Conjecture of Scannell and Sinha

Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in R n , for n ≤ 4. At the end of the paper they conjecture that when n is odd, the terms on the antidiagonal at the E 2 stage precisely give the space of irreducible Feynman diagrams related to the theory of Vassiliev invariants. In this paper we prove that conjecture. This has the application that the path components of the terms of the Taylor tower for the space of long knots in R 3 are in one-to-one correspondence with quotients of the module of Feynman diagrams, even though the Taylor tower does not actually converge. This provides strong evidence that the stages of the Taylor tower give rise to universal Vassiliev knot invariants in each degree. Our proof yields a sequence of new presentations for the space of irreducible Feynman diagrams.

Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S^3 and the similarity of the combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S^3 admitting lens space surgeries.

LINKING NUMBER IN A PROJECTIVE SPACE AS THE DEGREE OF A MAP

For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

A SURVEY OF BOTT–TAUBES INTEGRATION

It is well-known that certain combinations of configuration space integrals defined by Bott and Taubes [11] produce cohomology classes of spaces of knots. The literature surrounding this important fact, however, is somewhat incomplete and lacking in detail. The aim of this paper is to fill in the gaps as well as summarize the importance of these integrals.

SPINAL KNOTS IN LENS SPACES

A knot in a lens is said to be spinal if it can be isotoped on a standard spine (e.g. in ℝP3, spinal knots bound a Möbius band). We prove that a Dehn surgery on a non-spinal knot in a lens space cannot produce 𝕊3. With a view to study the Dehn surgeries that produce lens spaces, the main part is devoted to finding an obstruction for a standard spine to be minimal. We consider the intersection graphs coming from a standard spine and an arbitrary surface. This obstruction is given by the existence of a generalized Scharlemann cycle.

Hyperfinite knots via the CJKLS invariant in the thermodynamic limit

We set forth a definition of hyperfinite knots. Loosely speaking, these are limits of certain sequences of knots with increasing crossing number. These limits exist in appropriate closures of quotient spaces of knots. We give examples of hyperfinite knots. These examples stem from an application of the Thermodynamic Limit to the CJKLS invariant of knots.

TheC–polynomial of a knot

In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative A-polynomial of a knot. In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group U_q(SL_2)) specializes at q=1 to the better known A-polynomial of a knot, which has to do with genuine SL_2(C) representations of the knot complement. Computing the non-commutative A-polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the C-polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative A-polynomial of twist knots. Finally, we formulate a number of conjectures relating the A, the C-polynomial and the Alexander polynomial, all confirmed for the class of twist knots.