Prime Knots Research Articles

A first proof of knot localization for polymers in a nanochannel

Abstract Based on polymer scaling theory and numerical evidence, Orlandini, Tesi, Janse van Rensburg and Whittington conjectured in 1996 that the limiting entropy of knot-type K lattice polygons is the same as that for unknot polygons, and that the entropic critical exponent increases by one for each prime knot in the knot decomposition of K. This Knot Entropy (KE) conjecture is consistent with the idea that for unconfined polymers, knots occur in a localized way (the knotted part is relatively small compared to polymer length). For full confinement (to a sphere or box), numerical evidence suggests that knots are much less localized. Numerical evidence for nanochannel or tube confinement is mixed, depending on how the size of a knot is measured. Here we outline the proof that the KE conjecture holds for polygons in the ∞ x 2 x 1 lattice tube and show that knotting is localized when a connected-sum measure of knot size is used. Similar results are established for linked polygons. This is the first model for which the knot entropy conjecture has been proved.

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.

Factors of HOMFLY polynomials

We study factorizations of HOMFLY polynomials of certain prime knots and oriented links. We begin with a computer analysis of prime knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs.

Geometric learning of knot topology.

Knots are deeply entangled with every branch of science. One of the biggest open challenges in knot theory is to formalise a knot invariant that can unambiguously and efficiently distinguish any two knotted curves. Additionally, the conjecture that the geometrical embedding of a curve encodes information on its underlying topology is, albeit physically intuitive, far from proven. Here we attempt to tackle both these outstanding challenges by proposing a neural network (NN) approach that takes as input a geometric representation of a knotted curve and tries to make predictions of the curve's topology. Intriguingly, we discover that NNs trained with a so-called geometrical "local writhe" representation of a knot can distinguish curves that share one or many topological invariants and knot polynomials, such as mutant and composite knots, and can thus classify knotted curves more precisely than some knot polynomials. Additionally, we also show that our approach can be scaled up to classify all prime knots up to 10-crossings with more than 95% accuracy. Finally, we show that our NNs can also be trained to solve knot localisation problems on open and closed curves. Our main discovery is that the pattern of "local writhe" is a potentially unique geometric signature of the underlying topology of a curve. We hope that our results will suggest new methods for quantifying generic entanglements in soft matter and even inform new topological invariants.

Shortest paths of Rubik’s snake composite knots up to 8 crossings

A Rubik’s Snake is a toy that has been around for 40 years. It can be twisted to many interesting shapes. In particular a Rubik’s snake can be twisted to form a knot. In this paper we study how many blocks are needed to form a composite knot with up to 8 crossings. Also, we improved a couple of previous results for Rubik’s snake prime knots.

Tabulation of Knots Up to Five Triple-crossings and Moves between Oriented Diagrams

We enumerate and show tables of minimal diagrams for all prime knots up to the triple-crossing number equal to five. We derive a minimal generating set of oriented moves connecting triple-crossing diagrams of the same oriented knot. We also present a conjecture about a strict lower bound of the triple-crossing number of a knot related to the breadth of its Alexander polynomial.

On the triple point number of surface-links in Yoshikawa’s table

Yoshikawa made a table of knotted surfaces in [Formula: see text] with ch-index 10 or less. This remarkable table is the first to enumerate knotted surfaces analogous to the classical prime knot table. A broken sheet diagram of a surface-link is a generic projection of the surface in [Formula: see text] with crossing information along its singular set. The minimal number of triple points among all broken sheet diagrams representing a given surface-link is its triple point number. This paper compiles the known triple point numbers of the surface-links represented in Yoshikawa’s table and calculates or provides bounds on the triple point number of the remaining surface-links.

Distinguishing some genus one knots using finite quotients

We give a criterion for distinguishing a prime knot [Formula: see text] in [Formula: see text] from every other knot in [Formula: see text] using the finite quotients of [Formula: see text]. Using recent work of Baldwin–Sivek, we apply this criterion to the hyperbolic knots [Formula: see text], [Formula: see text], and the three-strand pretzel knots [Formula: see text] for every integer [Formula: see text].

The Ma–Qiu index and the Nakanishi index for a fibered knot are equal, and ω-solvability

For a knot [Formula: see text] in [Formula: see text], let [Formula: see text] be the knot group of [Formula: see text], [Formula: see text] the Ma–Qiu (MQ) index, which is the minimal number of normal generators of the commutator subgroup of [Formula: see text], and [Formula: see text] the Nakanishi index of [Formula: see text], which is the minimal number of generators of the Alexander module of [Formula: see text]. We generalize the notions for a pair of a group [Formula: see text] and its normal subgroup [Formula: see text], and we denote them by [Formula: see text] and [Formula: see text] respectively. Then it is easy to see [Formula: see text] in general. We also introduce a notion “[Formula: see text]-solvability” for a group that the intersection of all higher commutator subgroups is trivial. Our main theorem is that if [Formula: see text] is [Formula: see text]-solvable, then we have [Formula: see text]. As corollaries, for a fibered knot [Formula: see text], we have [Formula: see text], and we could determine the MQ indices of prime knots up to nine crossings completely.

Jones diameter and crossing number of knots

It has long been known that the quadratic term in the degree of the colored Jones polynomial of a knot is bounded above in terms of the crossing number of the knot. We show that this bound is sharp if and only if the knot is adequate.As an application of our result we determine the crossing numbers of broad families of non-adequate prime satellite knots. More specifically, we exhibit minimal crossing number diagrams for untwisted Whitehead doubles of zero-writhe adequate knots. This allows us to determine the crossing number of untwisted Whitehead doubles of any amphicheiral adequate knot, including, for instance, the Whitehead doubles of the connected sum of any alternating knot with its mirror image.We also determine the crossing number of the connected sum of any adequate knot with an untwisted Whitehead double of a zero-writhe adequate knot.

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.

A Cable Knot and BPS-Series

A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we observe a relation between the series invariant of the cable knot and the series invariant of the figure eight knot. This relation provides an alternative simple method of finding the former series invariant.

Regular projections of the link L6n1

Given a link projection [Formula: see text] and a link [Formula: see text], it is natural to ask whether it is possible that [Formula: see text] is a projection of [Formula: see text]. Taniyama answered this question for the cases in which [Formula: see text] is a prime knot or link with crossing number at most five. Recently, Takimura settled the issue for the knot [Formula: see text]. We answer this question for the case in which [Formula: see text] is the link [Formula: see text].

Completely stereospecific synthesis of a molecular cinquefoil (51) knot

<h2>Summary</h2> Although several molecular prime knots have been prepared to date, their stereoselective synthesis is still a rare occurrence. Here, we report a two-step, completely stereospecific preparation of a molecular cinquefoil (5₁) knot in 86% yield. The knotted product is generated from the covalent capture of a pentameric circular helicate. The absolute configuration of amino acid ester within the ligand strand directly determines the helical chirality of the circular helicate, the handedness, and the topological writhe of the knot. An open helicate features layered packing in the solid state, providing a scaffold for molecularly woven polymers. Several anions bind weakly to the well-organized central cavity of the knot through hydrogen bonding. Topological chirality alone induces all-important changes in the environment around the chromophores. Transfer of chirality in knot synthesis opens up opportunities for the assembly of other chiral nanotopologies and exploring the consequences of topological stereochemistry in chemistry, materials, and biology.

The smooth 4-genus of (the rest of) the prime knots through 12 crossings

We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.

Social Self-Sorting Synthesis of Molecular Knots.

We report the synthesis of molecular prime and composite knots by social self-sorting of 2,6-pyridinedicarboxamide (pdc) ligands of differing topicity and stereochemistry. Upon mixing achiral monotopic and ditopic pdc-ligand strands in a 1:1:1 ratio with Lu(III), a well-defined heteromeric complex featuring one of each ligand strand and the metal ion is selectively formed. Introducing point-chiral centers into the ligands leads to single-sense helical stereochemistry of the resulting coordination complex. Covalent capture of the entangled structure by ring-closing olefin metathesis then gives a socially self-sorted trefoil knot of single topological handedness. In a related manner, a heteromeric molecular granny knot (a six-crossing composite knot featuring two trefoil tangles of the same handedness) was assembled from social self-sorting of ditopic and tetratopic multi-pdc strands. A molecular square knot (a six-crossing composite knot of two trefoil tangles of opposite handedness) was assembled by social self-sorting of a ditopic pdc strand with four (S)-centers and a tetratopic strand with two (S)- and six (R)-centers. Each of the entangled structures was characterized by 1H and 13C NMR spectroscopy, mass spectrometry, and circular dichroism spectroscopy. The precise control of composition and topological chirality through social self-sorting enables the rapid assembly of well-defined sequences of entanglements for molecular knots.

New superbridge index calculations from non-minimal realizations

Previous work [C. Shonkwiler, New computations of the superbridge index, J. Knot Theory Ramifications 29(14) (2020) 2050096] used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. This work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygonal realizations do not minimize the stick number of the knot; these seem to be the first such examples. Appendix A gives a complete summary of what is currently known about superbridge indices of prime knots through 10 crossings and Appendix B gives all knots through 16 crossings for which the superbridge index is known.

Nontrivial Steenrod squares on prime, hyperbolic and satellite knots

We show that there are prime knots so that the Steenrod operations of Lipshitz and Sarkar arXiv:1204.5776 are non trivial on their Khovanov homology. This answers a question posed by Lipshitz and Sarkar in their paper arXiv:1709.03602. We then go on to show that there are hyperbolic and satellite knots so that the Steenrod operations are non trivial on their Khovanov homology.

Petal projections, knot colorings and determinants

An \"{u}bercrossing diagram is a knot diagram with only one crossing that may involve more than two strands of the knot. Such a diagram without any nested loops is called a petal projection. Every knot has a petal projection from which the knot can be recovered using a permutation that represents strand heights. Using this permutation, we give an algorithm that determines the $p$-colorability and the determinants of knots from their petal projections. In particular, we compute the determinants of all prime knots with crossing number less than $10$ from their petal permutations.

Infinitely many roots of unity are zeros of some Jones polynomials

Let \(N=2n^2-1\) or \(N=n^2+n-1\), for any \(n\ge 2\). Let \(M=\frac{N-1}{2}\). We construct families of prime knots with Jones polynomials \((-1)^M\sum _{k=-M}^{M} (-1)^kt^k\). Such polynomials have Mahler measure equal to 1. If N is prime, these are cyclotomic polynomials \(\Phi _{2N}(t)\), up to some shift in the powers of t. Otherwise, they are products of such polynomials, including \(\Phi _{2N}(t)\). In particular, all roots of unity \(\zeta _{2N}\) occur as roots of Jones polynomials. We also show that some roots of unity cannot be zeros of Jones polynomials.