Nontrivial Knot Research Articles

The Borel complexity of the space of left‐orderings, low‐dimensional topology, and dynamics

AbstractWe develop new tools to analyze the complexity of the conjugacy equivalence relation , whenever is a left‐orderable group. Our methods are used to demonstrate nonsmoothness of for certain groups of dynamical origin, such as certain amalgams constructed from Thompson's group . We also initiate a systematic analysis of , where is a 3‐manifold. We prove that if is not prime, then is a universal countable Borel equivalence relation, and show that in certain cases the complexity of is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of . We also prove that if is the complement of a nontrivial knot in then is not smooth, and show how determining smoothness of for all knot manifolds is related to the L‐space conjecture.

Lattice stick number 15 is unattainable for non-splittable links

In this paper, we explore mathematical links, defined as closed curves embedded in 3D space. Knot theory studies these structures, which also occur in real-world biopolymers like DNA. Lattice links are links in the cubic lattice. For scientific simulations or statistical studies, links are simplified to lattice links. The lattice stick number, denoted as s L (K), is the minimum number of lattice sticks needed to represent a link K in the cubic lattice. In previous study, it was shown that only two non-trivial knots and six non-splittable links have s L ≤ 14: specifically, sL(212)=8 , sL(31)=sL(212♯212) = sL(623)=sL(633)=12 , sL(412)=13 , and sL(41)=sL(512)=14 . Recent study has further revealed that no knot can have s L = 15. In this paper, we prove that lattice stick number 15 is not attainable for non-splittable links. As a corollary, eleven non-splittable links with s L =16 are presented.

Blanchfield pairings and Gordian distance

A lower bound of the Gordian distance is presented in terms of the Blanchfield pairing. Our approach, in particular, allows us to show at least for [Formula: see text] pairs of unoriented nontrivial prime knots with up to [Formula: see text] crossings that their Gordian distance is equal to [Formula: see text], most of which are difficult to treat otherwise.

Shortest paths of Rubik’s snake composite knots with 9 crossings

A Rubik’s Snake is an interesting toy that was invented over 40 years ago. It can be twisted to many interesting shapes, including non-trivial knots. In this paper we proposed a systematic way to study how many blocks are needed to form a composite knot with 9 crossings. We also improved some results from our previous papers.

Minimality of rational knots C(2n + 1,2m,2)

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots [Formula: see text] in the Conway notation, where [Formula: see text] and [Formula: see text] are integers. When [Formula: see text], we show that the nonabelian [Formula: see text]-character variety of [Formula: see text] is irreducible and therefore [Formula: see text] is a minimal knot. The proof of this result is an interesting application of Eisenstein’s irreducibility criterion for polynomials over integral domains.

Characterization of a spring pendulum phase-space trajectories.

We study the geometrical properties of phase-space trajectories (or orbits) of a spring pendulum as functions of the energy. Poincaré maps are used to describe the properties of the system. The points in the Poincaré maps of regular orbits (non-chaotic) cluster around separated segments or in chains of islands. Looking at how segments are formed, we conclude that the orbits are closely related to torus knots. Examining the toroidal and poloidal turns of the orbits, we introduce the definition of a rational parameter Ω, which is closely related to the concept of frequency used in the analysis of dynamical systems. Algorithms were developed to calculate Ω, and we found that this parameter naturally describes the orbits in terms of Farey sequences; also, calculations show that orbits with the same Ω have similar dynamics. Orbits corresponding to chains of islands are identified with cable knots that can be characterized using two parameters analogous to Ω. In some cases, non-trivial cable knots were found. With the analysis presented in this study, it is shown that Ω follows predictable distributions in the (z,Ω) space.

Discovery of a trefoil knot in the RydC RNA: Challenging previous notions of RNA topology

Knots are very common in polymers, including DNA and protein molecules. Yet, no genuine knot has been identified in natural RNA molecules to date. Upon re-examining experimentally determined RNA 3D structures, we discovered a trefoil knot 31, the most basic non-trivial knot, in the RydC RNA. This knotted RNA is a member of a small family of short bacterial RNAs, whose secondary structure is characterized by an H-type pseudoknot. Molecular dynamics simulations suggest a folding pathway of the RydC RNA that starts with a native twisted loop. Based on sequence analyses and computational RNA 3D structure predictions, we postulate that this trefoil knot is a conserved feature of all RydC-related RNAs. The first discovery of a knot in a natural RNA molecule introduces a novel perspective on RNA 3D structure formation and on fundamental research on the relationship between function and spatial structure of biopolymers.

Evolutionary Khovanov homology

<p>Knot theory, a subfield in geometric topology, is the study of the embedding of closed circles into three-dimensional Euclidean space, motivated by the ubiquity of knots in daily life and human civilization. However, focusing on topology, the current knot theory lacks metric analysis. As a result, the application of knot theory has remained largely primitive and qualitative. Motivated by the need of quantitative knot data analysis (KDA), this work implemented the evolutionary Khovanov homology (EKH) to facilitate a multiscale KDA of real-world data. EKH considers specific metrics to filter links, capturing multiscale topological features of knot configurations beyond traditional invariants. It is demonstrated that EKH can reveal non-trivial knot invariants at appropriate scales, even when the global topological structure of a knot is simple. The proposed EKH holds great potential for KDA and machine learning applications related to knot-type data, in contrast to other data forms, such as point cloud data and data on manifolds.</p>

Six Proofs of the Fáry–Milnor Theorem

We survey known proofs of the Fáry–Milnor theorem; it states that any nontrivial knot makes at least two full turns.

Exact hopfion vortices in a 3D Heisenberg ferromagnet

We find exact static soliton solutions for the unit spin vector field of an inhomogeneous, anisotropic three-dimensional Heisenberg ferromagnet. Each soliton is labeled by two integers n and m. It is a (modified) skyrmion in the z=0 plane with winding number n, which twists out of the plane m times in the z-direction to become a 3D soliton. Here m arises due to the periodic boundary condition at the z-boundaries. We use Whitehead's integral expression to find that the Hopf invariant of the soliton is an integer H=nm. It represents a closed hopfion vortex. Plots of the preimages of this topological soliton show that they are either unknots or nontrivial knots, depending on n and m. Any pair of preimage curves links H times, corroborating the interpretation of H as a linking number. We also calculate the exact energy of the hopfion vortex, and show that its topological lower bound has a sublinear dependence on H. Using Derrick's scaling analysis, we demonstrate that the presence of a spatial inhomogeneity in the anisotropic interaction, which in turn introduces a characteristic length scale in the system, leads to the stability of the hopfion vortex.

Cable knots obtained by positively twisting torus knots

The twisted torus knot [Formula: see text] is a knot obtained from the torus knot [Formula: see text] by twisting [Formula: see text] adjacent stands fully [Formula: see text] times. If [Formula: see text] or [Formula: see text] is a multiple of [Formula: see text], then [Formula: see text] is known to be a cable of a nontrivial knot or a torus knot. Assuming that [Formula: see text] and [Formula: see text] is not a multiple of [Formula: see text], we determine the parameters [Formula: see text] for which the positively twisted torus knot [Formula: see text] is a cable knot.

Null-homotopic knots have Property R

AbstractWe prove that if K is a nontrivial null-homotopic knot in a closed oriented 3–manfiold Y such that $Y-K$ does not have an $S^1\times S^2$ summand, then the zero surgery on K does not have an $S^1\times S^2$ summand. This generalises a result of Hom and Lidman, who proved the case when Y is an irreducible rational homology sphere.

Idempotents, free products and quandle coverings

In this paper, we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of quandles of finite type that are nontrivial coverings over nice base quandles admit infinitely many nontrivial idempotents, and give their complete description. We show that the set of all these idempotents forms a quandle in itself. As an application, we deduce that the quandle ring of the knot quandle of a nontrivial long knot admit nontrivial idempotents. We consider free products of quandles and prove that integral quandle rings of free quandles have only trivial idempotents, giving an infinite family of quandles with this property. We also give a description of idempotents in quandle rings of unions and certain twisted unions of quandles.

Upper and lower bounds on delta-crossing number and tabulation of knots up to four delta-crossings

In this paper, we will strengthen the known upper and lower bounds on the delta-crossing number of knots in terms of the triple-crossing number. The latter bound turns out to be strong enough to obtain unknown values of triple-crossing numbers for a few knots. We also prove that we can always find at least one tangle from a particular set of four tangles, in any triple-crossing projections of any non-trivial knot or non-split link. In the last section, we enumerate and generate tables of minimal delta-diagrams for all prime knots up to the delta-crossing number equal to four. We also give a concise survey about known inequalities between integer-valued classical knot invariants.

Fluid-electromagnetic helicities and knotted solutions of the fluid-electromagnetic equations

In this paper we consider an Euler fluid coupled to external electromagnetism. We prove that the Hopfion fluid-electromagnetic knot, carrying fluid and electromagnetic (EM) helicities, solves the fluid dynamical equations as well as the Abanov Wiegmann (AW) equations for helicities, which are inspired by the axial-current anomaly of a Dirac fermion. We also find a nontrivial knot solution with truly interacting fluid and electromagnetic fields. The key ingredients of these phenomena are the EM and fluid helicities. An EM dual system, with a magnetically charged fluid, is proposed and the analogs of the AW equations are written down. We consider a fluid coupled to a nonlinear generalizations for electromagnetism. The Hopfions are shown to be solutions of the generalized equations. We write down the formalism of fluids in 2+1 dimensions, and we dimensionally reduce the 3+1 dimensional solutions. We determine the EM knotted solutions, from which we derive the fluid knots, by applying special conformal transformations with imaginary parameters on un-knotted null constant EM fields.

Instanton L-spaces and splicing

We prove that the 3-manifold obtained by gluing the complements of two nontrivial knots in homology 3-sphere instanton L-spaces, by a map which identifies meridians with Seifert longitudes, cannot be an instanton L-space. This recovers the recent theorem of Lidman–Pinzón-Caicedo–Zentner that the fundamental group of every closed, oriented, toroidal 3-manifold admits a nontrivial SU(2)-representation, and consequently Zentner’s earlier result that the fundamental group of every closed, oriented 3-manifold besides the 3-sphere admits a nontrivial SL(2,ℂ)-representation.

Knots admitting purely cosmetic surgeries are prime

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in S3 do not admit purely cosmetic surgeries. In this article, we show that knots admitting purely cosmetic surgeries are prime.

Instantons and L-space surgeries

We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the \operatorname{Spin}^c decompositions of cobordism maps in other Floer homology theories. As our main application, we prove (modulo a mild nondegeneracy condition) that for r a positive rational number and K a nontrivial knot in the 3 -sphere, there exists an irreducible homomorphism \pi_1(S^3_r(K)) \to SU(2) unless r \geq 2g(K)-1 and K is both fibered and strongly quasipositive, broadly generalizing results of Kronheimer and Mrowka. We also answer a question of theirs from 2004, proving that there is always an irreducible homomorphism from the fundamental group of 4-surgery on a nontrivial knot to SU(2) . In another application, we show that a slight enhancement of the A -polynomial detects infinitely many torus knots, including the trefoil.

Vertex distortion detects the unknot

The first two authors introduced vertex distortion of lattice knots and showed that the vertex distortion of the unknot is 1. It was conjectured that the vertex distortion of a knot class is 1 if and only if it is trivial. We use Denne and Sullivan’s lower bound on Gromov distortion to bound the vertex distortion of non-trivial lattice knots. This bounding allows us to conclude that a knot class has vertex distortion 1 if and only if it is trivial. We also show that vertex distortion does not have a universal upper bound and provide a vertex distortion calculator.

Enumerating Nontrivial Knot Mosaics with SAT (Student Abstract)

Mathematical knots are interesting topological objects. Using simple arcs, lines, and crossings drawn on eleven possible tiles, knot mosaics are a representation of knots on a mosaic board. Our contribution is using SAT solvers as a tool for enumerating nontrivial knot mosaics. By encoding constraints for local knot mosaic properties, we computationally reduce the search space by factors of up to 6600. Our future research directions include encoding constraints for global properties and using parallel SAT techniques to attack larger boards.