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In The Unseen Universe, authors Tait and Balfour Stewart hypothesized about the nature of the universe. They postulated that just as a smoke ring is composed of ordinary molecules, so too those molecules might be vortex rings of "something much finer and more subtle than themselves." In this sketch, the innermost circle represents a smoke ring within our visible universe (1). In the same way, our universe is an impermanent part of a more perfect universe (2), which is itself part of an even more perfect universe (3), and so on, until reaching a universe with "infinite energy" created by a divine agent.
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The present work proposes a methodology for failure analysis using knot theory.
Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian retains the topological invariants of Khovanov homology, while its non-harmonic spectrum reveals additional informatio...
We report on our success on solving the problem of untangling braids up to length 20 and width 4. We use feed-forward neural networks in the framework of reinforcement learning to train the agent to choose Reidemeister moves to untangle braids in the minimal number of moves.
The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagr...
The art of tying knots is exploited in nature and occurs in multiple applications ranging from being an essential part of scouting programs to engineering molecular knots. Biomolecular knots, such as knotted proteins, bear various cellular functions and their entanglement is believed to provide them with thermal and kinetic stability. Yet, little i...
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... The history of Knot theory starts from 19th century physics, with the work of Gauss on computing linking numbers in a system of linked circular wires. D. Silver [7] also studied the knots and coined the word topology. In 1867, Kelvin's vortex model of atom by W.T Thompson [8] was presented. ...
Knot theory is the Mathematical study of knots. In this paper we have studied the Composition of two knots. Knot theory belongs to Mathematical field of Topology, where the topological concepts such as topological spaces, homeomorphisms, and homology are considered. We have studied the basics of knot theory, with special focus on Composition of knots, and knot determinants using Alexander Polynomials. And we have introduced the techniques to generalize the solution of composition of knots to present how knot determinants behave when we compose two knots.
... Vortex rings have been also investigated through experiments [15][16][17][18][19][20] . As reported in Silver 21 , the construction of a first "smoke cannon machine" is attributed to Peter Guthrie Tait. He constructed a wooden box with a circular hole carved at one end, then he covered the other end with a tightly stretched towel and placed ammonia and sulfuric acid inside the box. ...
Vortex rings can easily be generated in the laboratory or with homemade devices, but they have also been observed on volcanoes, since the eighteenth century. However, the physical conditions under which volcanic vortex rings form are still unknown. In order to better understand this phenomenon and provide clues on the dynamics of the volcanic vortex rings, we performed a series of finite element simulations to investigate which model configuration leads to the rings formation that best matches the field observations. Results show that the formation of volcanic vortex rings requires a combination of fast gas release from gas bubbles (slugs) at the top of the magma conduit and regularity in the shape of the emitting vent. Our findings offer important insights into the geometry of the uppermost portion of vortex-forming volcanic conduits. Volcanic vortex ring studies may form the basis for a cross-disciplinary assessment of the upper conduit dynamics of volcanic vents.
... In studies of topological solitons and singular defects, such classifications provide a means of summarizing topologically different structures in the order parameter fields, although the existence of a nontrivial element in the homotopy class does not guarantee their energetic stability or experimental observation [55]. The n-dimensional spheres (n-spheres, denoted as n ) are defined as sets of points equidistant from the origin in n+1 dimension, with an 1 circle being the 1-sphere embedded in 2D space (ℝ 2 ), 2 being an ordinary sphere embedded in 3D space (ℝ 3 ) and 0 being 0-sphere embedded in ℝ 1 that comprises 2 points equidistant from the origin in 1D, and so on [56][57][58][59][60][61][62]. The homotopy group labelled as πi( n ) is the i-th homotopy group that enlists the topologically different maps from i into n , where none of the distinct mappings can be continuously deformed 13 to the other mappings ( Fig. 2) [22, 53,55,61,62]. ...
Widely known for their uses in displays and electro-optics, liquid crystals are more than just technological marvels. They vividly reveal the topology and structure of various solitonic and singular field configurations, often markedly resembling the ones arising in many field theories and in the areas ranging from particle physics to optics, hard condensed matter and cosmology. In this review, we focus on chiral nematic liquid crystals to show how these experimentally highly accessible systems provide valuable insights into the structure and behavior of fractional, full, and multi-integer two-dimensional skyrmions, dislocations and both abelian and non-abelian defect lines, as well as various three-dimensionally localized, often knotted structures that include hopfions, heliknotons, torons and twistions. We provide comparisons of some of these field configurations with their topological counterparts in chiral magnets, discussing close analogies between these two condensed matter systems.
... In addition to the results reported here, regarding DNA and quantum gravity, contributions of knot theory in relation to certain fish species, polymer chains or telephone network architecture (Murasugi 1996) 9 are already recorded. All these have convinced today"s mathematicians that the knots are basic constructs, indispensable concepts, like the circle, essentially defining a fundamental relationship between certain quantities (Silver 2006). They reveal the extraordinary connections existing between the topological constructions regarding certain bodies existing in the macrocosm and some subtle aspects existing in the microcosm, in the fundamental constitutions of the physical world or in the architecture of the essential bricks of life (Francoise et al 2006, 220). ...
The dialogue between theology and science, nowadays more and more frequently approached from different perspectives in the Romanian space, proves relevant, among other things, in establishing a way of receiving scientific data about the world. A whole series of important discoveries that were made in the last century has revealed new data regarding the structure of space-time, several quantum constituents of matter and energy, unexpected regularities present in the living world and in the processes of the physical world. Such data have elicited long debates and led to the formulation of several questions such as those regarding the nature of the human mind, the nature of mathematical constructions and how the structures of the living and nonliving world can, to a large extent and in many respects, be rendered in mathematical terms.
As scientific exploration reveals more and more situations in which certain mathematical concepts and constructs prove adequate to describe some particular aspects of physical reality, new nuances and possible answers are added to the old questions, further increasing the consistency of discussions and formulation of nuanced perspectives. The order, the harmonies, the patterns existing in the structures and processes of the physical world are most easily located in the space of dialogue between theology, philosophy and science, as they bring into question both the problem of the mind, of the mathematical instrument, the question of the fact of science, but also the remarkable situation of the convergence of three aspects, namely matching the mind and its mathematics with the structures of the surrounding reality. Such situations naturally require a triple perspective, a correct use of the data of science and its premises, a philosophical contribution meant, among other things, to provide possible appropriate interpretations and understandings and, finally, a more comprehensive theological approach, capable of proposing meanings that link these contents with the life of man, with his spiritual uplifting.
We shall next highlight, through three recent work areas, some results that highlight three possible places of convergence between theology, philosophy and science, mentioning, where appropriate, important aspects that condition the authentic dialogue.
... Topology, like many fields of mathematics, owes some of its early development to questions arising in physics. A classic example of this is the development of knot theory: Lord Kelvin and Tait, inspired by experiments of Helmholtz, theorized that atoms were knotted tubes of aether, distinguished by their knot type [43]. This theory was quickly dismissed, but Tait's tabulation of knot projections with few crossings is arguably the dawn of modern knot theory. ...
The necklace braid group is the motion group of the n+1 component necklace link in Euclidean . Here consists of n pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group , especially those obtained as extensions of representations of the braid group and the loop braid group . We show that any irreducible representation extends to in a standard way. We also find some non-standard extensions of several well-known -representations such as the Burau and LKB representations. Moreover, we prove that any local representation of (i.e., coming from a braided vector space) can be extended to , in contrast to the situation with . We also discuss some directions for future study from categorical and physical perspectives.
... Although ultimately discarded, the theory did lead Peter Guthrie Tait and James Clerk Maxwell to begin working on the mathematics of knots, providing a nucleus for further studies in knot theory. 2 Knots and interlocked links remained a mathematical curiosity outside the purview of chemists until the first synthesis and isolation of a molecular Hopf linka [2]catenane-in 1960. 3 However, it wasn't until almost 30 years later, after the advent of metal-ion template approaches, that the first molecular knot-a trefoil knot ( Figure 1B)-was synthesized. ...
... For example, sodium, with its two spectral lines, could be considered as two interlocked rings-a Hopf link ( Figure 1A). Although ultimately discarded, the theory did lead Peter Guthrie Tait and James Clerk Maxwell to begin working on the mathematics of knots, providing a nucleus for further studies in knot theory. 2 Knots and interlocked links remained a mathematical curiosity outside the purview of chemists until the first synthesis and isolation of a molecular Hopf linka [2]catenane-in 1960. 3 However, it wasn't until almost 30 years later, after the advent of metal-ion template approaches, that the first molecular knot-a trefoil knot ( Figure 1B)-was synthesized. ...
In this issue of Chem, Fujita and co-workers investigate the self-assembly of peptide-based ligands with silver ions. Through selective crystallization they were able to isolate a molecular septafoil knot and a quadruply braided [2]catenane (8²1 link)—entangled molecules that had, until now, evaded realization.
... Their important roles in fluid dynamics and magnetohydrodynamics have long been recognized within the realm of classical physics [1]. The topological invariants describing these structures fascinated Gauss, Maxwell, and Kelvin [2] long before the concepts of knots and braids took a strong hold in topological quantum field theory [3]. In this paper, we investigate the topological structures hidden in the momentumtime space for quantum systems driven out of equilibrium. ...
... (1) The 4-zero terms (F = 0) and 3-zero terms (F = 1) vanish due to the anti-symmetric Levi-Civita symbol; (2) For the 2-zero terms (F = 2), the trace requires the other two nonzero elements to be the same; (3) For F = 4, the trace requires two of them (and the other two) to be equal. For cases (2)(3), the contributions to W 3 vanish due to the anti-symmetric Levi-Civita symbol. Now let us calculate the trace for the 1-zero terms with F = 3. ...
The quench dynamics following a sudden change in the Hamiltonian of a quantum system can be very complex. For band insulators, the global properties of a quantum quench may be captured by its topological invariants over the space-time continuum, which are intimately related to the static band topology of the pre- and post-quench Hamiltonian. We introduce the concept of loop unitary and its homotopy invariant to fully characterize the quench dynamics of arbitrary two-band insulators in two dimensions, going beyond existing scheme based on Hopf invariant which is only valid for trivial initial states. The theory traces the origin of nontrivial dynamical topology to the emergence of -defects in the phase band of , and establishes that , i.e. the Chern number change across the quench. We further show that the dynamical singularity is also encoded in the winding of the eigenvectors of along a lower dimensional curve where dynamical quantum phase transition occurs, if the pre- or post-quench Hamiltonian is trivial. The winding along this curve is related to the Hopf link, and gives rise to torus links and knots for quench to Hamiltonians with Dirac points. This framework can be generalized to multiband systems and other dimensions.
... Although it all started with Kelvin's hypothesis that different elements are differently knotted vortices of ether [20], most of the work on knot recognition is a typical mathematical endeavour, with little care paid to potential applications. Nevertheless, current real life motivation to study knots exist: for example, some molecules, such as DNA, can occur knotted, and their chemical or biological properties depend on the way they are knotted (it is reported that certain antibiotics exploit topological properties of DNA). ...
This is a report on our ongoing research on a combinatorial approach to knot
recognition, using coloring of knots by certain algebraic objects called
quandles. The aim of the paper is to summarize the mathematical theory of knot
coloring in a compact, accessible manner, and to show how to use it for
computational purposes. In particular, we address how to determine colorability
of a knot, and propose to use SAT solving to search for colorings. The
computational complexity of the problem, both in theory and in our
implementation, is discussed. In the last part, we explain how coloring can be
utilized in knot recognition.
... The topology of knots has been studied for almost two centuries. The earliest work dates back to the 19th century [2,3], when P. G. Tait first proposed the current classification of knotted loops based on the crossings observed in their twodimensional projection. Since then, knot theory has been applied to a wide number of areas in physics and beyond: of particular relevance to our work are the applications to biophysics, as several biopolymers are observed to form knots. ...
We investigate the sedimentation of knotted polymers by means of stochastic rotation dynamics, a molecular dynamics algorithm that takes hydrodynamics fully into account. We show that the sedimentation coefficient s, related to the terminal velocity of the knotted polymers, increases linearly with the average crossing number n_{c} of the corresponding ideal knot. This provides direct computational confirmation of this relation, postulated on the basis of sedimentation experiments by Rybenkov et al. [J. Mol. Biol. 267, 299 (1997)]. Such a relation was previously shown to hold with simulations for knot electrophoresis. We also show that there is an accurate linear dependence of s on the inverse of the radius of gyration R_{g}^{-1}, more specifically with the inverse of the R_{g} component that is perpendicular to the direction along which the polymer sediments. When the polymer sediments in a slab, the walls affect the results appreciably. However, R_{g}^{-1} remains to a good precision linearly dependent on n_{c}. Therefore, R_{g}^{-1} is a good measure of a knot's complexity.
... Suppose that tangles O b , R satisfy the following equations:(i) N (O b ) =< 1 > (substrate = unknot) (ii) N (O b + R) =< 2 > (1 st round product = Hopf link, with linking number = −1) (iii) N (O b + 2R) =< 2, 1,1 > (2 nd round product = Fig. 8 knot).Then{O b , R} = {(−3, 0),(1)} and the result of 3 rounds of processive recombination is the +Whitehead link ...
Cellular DNA is a long, thread-like molecule with remarkably complex
topology. Enzymes that manipulate the geometry and topology of cellular
DNA perform many vital cellular processes (including segregation of
daughter chromosomes, gene regulation, DNA repair, and generation of
antibody diversity). Some enzymes pass DNA through itself via
enzyme-bridged transient breaks in the DNA; other enzymes break the DNA
apart and reconnect it to different ends. In the topological approach to
enzymology, circular DNA is incubated with an enzyme, producing an
enzyme signature in the form of DNA knots and links. By observing the
changes in DNA geometry (supercoiling) and topology (knotting and
linking) due to enzyme action, the enzyme binding and mechanism can
often be characterized. This paper will discuss some personal research
history, and the tangle model for the analysis of site-specific
recombination experiments on circular DNA.
