Tame Knot Research Articles

Cyclic coverings of the 3-sphere branched over wild knots of dynamically defined type

Let [Formula: see text] be a tame knot and consider an [Formula: see text] beaded necklace [Formula: see text] which is the union of [Formula: see text] consecutive disjoint closed round balls (pearls) [Formula: see text], [Formula: see text]. An [Formula: see text] pearl chain necklace [Formula: see text] is the union of [Formula: see text] and [Formula: see text]. We will construct, via the action of a Kleinian group, a sequence of nested pearl chain necklaces [Formula: see text] whose inverse limit is a wild knot of dynamically defined type [Formula: see text]. In this paper, we will prove some topological properties of this kind of wild knots; in particular, we generalize the construction of cyclic branched coverings for this case, and we show that there exists a wild knot of dynamically defined type such that [Formula: see text] is an [Formula: see text]-fold cyclic branched covering of [Formula: see text] along it, for [Formula: see text].

Zero Set Structure of Real Analytic Beltrami Fields

In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in {mathbb {R}}^3, there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue +1 of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT

We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem.

Free quandles and knot quandles are residually finite

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core, and Alexander quandles) is established. Further, residual finiteness of automorphism groups of some residually finite quandles is also discussed.

Cubulated moves for 2-knots

In this paper, we prove that given two cubical links of dimension two in [Formula: see text], they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots (links) of dimensions one and two, respectively.

Knot mosaic tabulation

In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tame knots. In this article, we consider the mosaic number of a knot which is a natural and fundamental knot invariant defined in the knot mosaic system. We determine the mosaic number for all eight-crossing or fewer prime knots. This work is written at an introductory level to encourage other undergraduates to understand and explore this topic. No prior of knot theory is assumed or required.

Complexity of a knot invariant

AbstractThe algebraic structures called quandles constitute a complete invariant for tame knots. However, determining when two quandles are isomorphic is an empirically hard problem, so there is some dissatisfaction with quandles as knot invariants. We have confirmed this apparent difficulty, showing within the framework of Borel reducibility that the general isomorphism problem for quandles is as complex as possible. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computing fundamental groups from point clouds

We describe an algorithm for computing a finite, and typically small, presentation of the fundamental group of a finite regular CW-space. The algorithm is based on the construction of a discrete vector field on the $$3$$3-skeleton of the space. A variant yields the homomorphism of fundamental groups induced by a cellular map of spaces. We illustrate how the algorithm can be used to infer information about the fundamental group $$\pi _1(K)$$?1(K) of a metric space $$K$$K using only a finite point cloud $$X$$X sampled from the space. In the special case where $$K$$K is a $$d$$d-dimensional compact manifold $$K\subset \mathbb R^d$$K?Rd, we consider the closure of the complement of $$K$$K in the $$d$$d-sphere $$M_K=\overline{\mathbb S^d\!\setminus \!K}$$MK=Sd\K¯. For a base-point $$x$$x in the boundary $$\partial M_K$$?MK of the manifold $$M_K$$MK one can attempt to determine, from the point cloud $$X$$X, the induced homomorphism of fundamental groups $$\phi :\pi _1(\partial M_K,x)\rightarrow \pi _1(M_K,x)$$?:?1(?MK,x)??1(MK,x) in the category of finitely presented groups. We illustrate a computer implementation for $$K$$K a small closed tubular neighbourhood of a tame knot in $$\mathbb R^3$$R3. In this case the homomorphism $$\phi $$? is known to be a complete ambient isotopy invariant of the knot. We observe that low-index subgroups of finitely presented groups provide useful invariants of $$\phi $$?. In particular, the first integral homology of subgroups $$G < \pi _1(M_K)$$G<?1(MK) of index at most six suffices to distinguish between all prime knots with 11 or fewer crossings (ignoring chirality). We plan to provide formal time estimates for our algorithm and characteristics of a high performance C++ implementation in a subsequent paper. The prototype computer implementation of the present paper has been written in the interpreted gap programming language for computational algebra.

On some open 3-manifolds that are branched coverings of the Poincaré homology sphere

Blankinship and Fox proved that the Whitehead’s contractible, open \(3\)-manifold \(W\) is not an open \(3\)-cell by constructing a representation \(\rho \) of \(\pi _{1}(W\backslash \alpha )\) into the symmetric group of \(5\) elements \(\Sigma _{5}\), where \(\alpha \) is a particular tame knot in \(W\). A geometric explanation of this fact is given by showing that \(-1\) Dehn surgery on \(\alpha \) is a branched covering over the Poincare homology sphere. Similar constructions exist for the complement in \(S^{3}\) of a Cantor set constructed by Bing, leading to the conjectured existence of a branched covering \(p:M\rightarrow S^{3}\) of a \(3\)-manifold \(M\), branched over a (wildly embedded) Cantor set.

Knottedness is in [formula omitted], modulo GRH

Given a tame knot K presented in the form of a knot diagram, we show that the problem of determining whether K is knotted is in the complexity class NP, assuming the generalized Riemann hypothesis (GRH). In other words, there exists a polynomial-length certificate that can be verified in polynomial time to prove that K is non-trivial. GRH is not needed to believe the certificate, but only to find a short certificate. This result complements the result of Hass, Lagarias, and Pippenger that unknottedness is in NP. Our proof is a corollary of major results of others in algebraic geometry and geometric topology.

THE LOMONACO–KAUFFMAN CONJECTURE

Samuel J. Lomonaco Jr. and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco–Kauffman conjecture.

CUBULATED MOVES AND DISCRETE KNOTS

In this paper, we prove that two cubic knots K1, K2 are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use this fact to describe a cubic knot in a discrete way, as a cyclic permutation of contiguous vertices of the ℤ3-lattice (with some restrictions); moreover, we describe a regular diagram of a cubic knot in terms of such cyclic permutations.

AN INFINITE FAMILY OF KNOTS WHOSE MOSAIC NUMBER IS REALIZED IN NON-REDUCED PROJECTIONS

Lomonaco and Kauffman [Quantum knots and mosaics, Quantum Inf. Process. 7(2–3) (2008) 85–115] introduced the notion of knot mosaics in their work on quantum knots. It is conjectured that knot mosaic type is a complete invariant of tame knots. In this paper, we answer a question of C. Adams by constructing an infinite family of knots whose mosaic number can be realized only when the crossing number is not. That is, there is an infinite family of non-minimal knots whose mosaic numbers are known.

Mirror-curves and knot mosaics

Inspired by the paper on quantum knots and knot mosaics (Lomonaco and Kauffman, 2008 [18]) and grid diagrams (or arc presentations), used extensively in the computations of Heegaard–Floer knot homology (Bar-Natan, 0000 [16], Cromwell, 1995 [21], Manolescu et al., 2007 [22]), we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot mosaics (Lomonaco and Kauffman, 2008 [18]), mirror-curves, and grid diagrams (Bar-Natan, 0000 [16], Cromwell, 1995 [21], Kuriya, 2008 [20], Manolescu et al., 2007 [22]). Hence, we introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grids, suitable for software implementation. We provide tables of minimal mirror-curve codes for knots and links obtained from rectangular grids of size 3×3 and p×2 (p≤4), and describe an efficient algorithm for computing the Kauffman bracket and L-polynomials (Jablan and Sazdanović, 2007 [8], Kauffman, 2006 [11], Kauffman, 1987 [12]) directly from mirror-curve representations.

The distortion of a knotted curve

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed that any closed curve has distortion at least π/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5π/3; examples show that distortion under 7.16 suffices to build a trefoil knot. Our argument uses the existence of a shortest essential secant and a characterization of borderline-essential arcs.

Quantum knots and mosaics

In this paper, we give a precise and workable definition of a knot system, the states of which are called knots. This definition can be viewed as a blueprint for the construction of an actual physical system. Moreover, this definition of a knot system is intended to represent the quantum embodiment of a closed knotted physical piece of rope. A knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a knot can exhibit non-classical behavior, such as superposition and entanglement. This raises some interesting and puzzling questions about the relation between topological and entanglement. The knot type of a knot is simply the orbit of the knot under the action of the ambient group. We investigate observables which are invariants of knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.

The diameter of the set of boundary slopes of a knot

Let K be a tame knot with irreducible exterior M(K) in a closed, connected, orientable 3--manifold Sigma such that pi_1(Sigma) is cyclic. If infinity is not a strict boundary slope, then the diameter of the set of strict boundary slopes of K, denoted d_K, is a numerical invariant of K. We show that either (i) d_K >= 2 or (ii) K is a generalized iterated torus knot. The proof combines results from Culler and Shalen [Comment. Math. Helv. 74 (1999) 530-547] with a result about the effect of cabling on boundary slopes.

RECURSION FORMULAS FOR SOME ABELIAN KNOT INVARIANTS

Let K be a tame knot in S3. We show that the sequence of cyclic resultants of the Alexander polynomial of K satisfies a linear recursion formula with integral coefficients. This means that the orders of the first homology groups of the branched cyclic covers of K can be computed recursively. We further establish the existence of a recursion formula that generates sequences which contain the square roots of the orders for the odd-fold covers that contain the square roots of the orders for the even-fold covers quotiented by the order for the two-fold cover. (That these square roots are all integers follows from a theorem of Plans.)

The Curvature of Lattice Knots

A result of Milnor [1] states that the infimum of the total curvature of a tame knot K is given by 2πμ(K), where μ(K) is the crookedness of the knot K. It is also known that μ(K)=b(K), where b(K) is the bridge index of K [2]. The situation appears to be quite different for knots realised as polygons in the cubic lattice. We study the total curvature of lattice knots by developing algebraic techniques to estimate minimal curvature in the cubic lattice. We perform simulations to estimte the minimal curvature of lattice knots, and conclude that the situation is very different than for tame knots in ℛ3.

Flows on ³ supporting all links as orbits

We construct counterexamples to some conjectures of J. Birman and R. F. Williams concerning the knotting and linking of closed orbits of flows on 3-manifolds. By establishing the existence of “universal templates,” we produce examples of flows on S3 containing closed orbits of all knot and link types simultaneously. In particular, the set of closed orbits of any flow transverse to a fibration of the complement of the figure-eight knot in S3 over S1 contains representatives of every (tame) knot and link isotopy class. Our methods involve semiflows on branched 2-manifolds, or templates. In this announcement, we answer some questions raised by Birman and Williams in their original examination of the link of closed orbits in the flow on S induced by the fibration of the complement of a fibred knot or link [4]. In their work, they proposed the following conjecture: Conjecture 0.1 (Birman and Williams, 1983).The figure-eight knot (K8) does not appear as a closed orbit of the flow induced by the fibration of the complement of the figure-eight knot. By “the” flow is meant the flow obtained by integrating the gradient of p, where p : S \K8 → S is the unique fibration over S whose monodromy is the pseudoAnosov representative of its isotopy class, with respect to the Nielsen-Thurston classification [16, 9]. We have resolved this conjecture in the negative and, in so doing, have discovered interesting examples of flows on 3-manifolds and semiflows on branched 2-manifolds. More specifically, in Theorem 3.1, we show that any fibration of the complement of the figure-eight knot in S over S induces a flow on S containing every tame knot and link as closed orbits. We include only those definitions and results which are relevant for this announcement, leaving details to a separate work [11]. 1. Template Theory Periodic orbits of a flow are embedded circles. When the flow is three-dimensional, periodic orbits are knots and the collection of periodic orbits forms a link which often is nontrivial. A valuable tool for examining knotted periodic orbits in three dimensional flows is the template construction of Birman and Williams [3, 4]. Received by the editors June 16, 1995. 1991 Mathematics Subject Classification. Primary 57M25, 58F22; Secondary 58F25, 34C35.