Loops, knots, strings,..

Knots are ubiquitous: We encounter them in woven materials, in the numerous sailors’ knots, and in constantly entangled electric cables and extension cords. When putting on their shoes, even small children learn to master their first knots – long before they can read and write. Even our DNA can be complicatedly knotted. From a mathematical point of view, knots that seem completely different at first sight can belong to the same class. The essential criterion is that one knot can be transformed into another by means of simple deformations. The most simple example is a rubber band. Topologically speaking, every shape you can create from it without cutting open the loop and joining it back together is equivalent to the initial rubber band. A completely different knot, for example, is the trefoil knot (see figure 1). This knot cannot easily be tied from an intact rubber band. Furthermore, several interlocked loops can constitute even more complex structures.

TGD can much be seen as a Universe of knots, loops or bubbles. See Knots in TGD. Also E. Witten. Fivebranes and Knots.

Knots can now be tied systematically in the microscopic world. A team at the Max Planck Institute for Dynamics and Self-Organization in Gottingen (Germany) since September 2010, has now found a way to create every imaginable knot inside a liquid crystal. Starting points of the new method are tiny silica microspheres confined in thin liquid crystal layers. Surrounding these microspheres, a net of fine lines is formed where the molecular orientation of the liquid crystal is altered. The researchers discovered a method to twist and link these lines in such a manner as to create every knot imaginable.

A single silica microsphere entering the layer changes the surrounding alignment substantially: around the sphere a ring-shaped region forms in which no preferred direction can be discerned. Scientists refer to such disruptions in the molecular order as defect lines. Since the defect ring surrounding a microsphere reflects light differently than the rest of the liquid crystal, it can be easily detected. "It looks as if every microsphere were surrounded by its own ring – similar to the planet Saturn", explains Tkalec . These Saturn’s rings are oriented perpendicularly to the average orientation of molecules between the glass plates.

Creating knots requires the defect rings of neighbouring microspheres to be able to attach to each other in two directions. In order to achieve this, the scientists used a "trick": if the upper plate confining the liquid crystal layer is turned by 90 degrees, the alignment of the molecules is changed. While the lower molecules still point in the same direction as before, the upper ones are also rotated by 90 degrees. In between, the transition is gradual. Scientists refer to this as a twisted or chiral nematic liquid crystal. "In this experimental set-up, the defect rings surrounding the spheres are slightly buckled - like a buckled wheel of a bicycle".

Uroš Tkalec, Miha Ravnik, Simon Čopar, Slobodan Žumer und Igor Muševič, Reconfigurable Knots and Links in Chiral Nematic Colloids, Science, 1 July 2011. http://www.science … /62.abstract

Tying knots and linking microscopic loops of polymers, macromolecules, or defect lines in complex materials is a challenging task for material scientists. We demonstrate the knotting of microscopic topological defect lines in chiral nematic liquid-crystal colloids into knots and links of arbitrary complexity by using laser tweezers as a micromanipulation tool. All knots and links with up to six crossings, including the Hopf link, the Star of David, and the Borromean rings, are demonstrated, stabilizing colloidal particles into an unusual soft matter. The knots in chiral nematic colloids are classified by the quantized self-linking number, a direct measure of the geometric, or Berry’s, phase. Forming arbitrary microscopic knots and links in chiral nematic colloids is a demonstration of how relevant the topology can be for the material engineering of soft matter.

Provided by Max-Planck-Gesellschaft (news : web)

A revolution in knot theory,
Figure 1. This knot has Gauss code O1U2O3U1O2U3. Graphic by Sam Nelson. As Nelson reports, mathematicians have devised various ways to represent the information contained in knot diagrams. One example is the Gauss code, which is a sequence of letters and numbers wherein each crossing in the knot is assigned a number and the letter O or U, depending on whether the crossing goes over or under.

In his article “The Combinatorial Revolution in Knot Theory”, to appear in the December 2011 issue of the Notices of the AMS, Sam Nelson describes a novel approach to knot theory that has gained currency in the past several years and the mysterious new knot-like objects discovered in the process.

Knot theory is usually understood to be the study of embeddings of topological spaces in other topological spaces. Classical knot theory, in particular, is concerned with the ways in which a circle or a disjoint union of circles can be embedded in R3. Knots are usually described via knot diagrams, projections of the knot onto a plane with breaks at crossing points to indicate which strand passes over and which passes under, as in Figure 1. However, much as the concept of “numbers” has evolved over time from its original meaning of cardinalities of finite sets to include ratios, equivalence classes of rational Cauchy sequences, roots of polynomials, and more, the classical concept of “knots” has recently undergone its own evolutionary generalization. Instead of thinking of knots topologically as ambient isotopy classes of embedded circles or geometrically as simple closed curves in R3, a new approach defines knots combinatorially as equivalence classes of knot diagrams under an equivalence relation determined by certain diagrammatic moves. No longer merelysymbols standing in for topological or geometric objects, the knot diagrams themselves have become mathematical objects of interest.
Identifying knot invariants (functions used to distinguish different knot types) by checking invariance under the moves has been common ever since. A recent shift toward taking the combinatorial approach more seriously, however, has led to the discovery of new types of generalized knots and links that do not correspond to simple closed curves in R3.

Like the complex numbers arising from missing roots of real polynomials, the new generalized knot types appear as abstract solutions in knot equations that have no solutions among the classical geometric knots. Although they seem esoteric at first, these generalized knots turn out to have interpretations such as knotted circles or graphs in three-manifolds other than R3, circuit diagrams, and operators in exotic algebras. Moreover, classical knot theory emerges as a special case of the new generalized knot theory. This diagram-based combinatorial approach to knot theory has revived interest in a related approach to algebraic knot invariants, applying techniques from universal algebra to turn the combinatorial structures into algebraic ones. The resulting algebraic objects,with names such as kei, quandles, racks, and biquandles, yield new invariants of both classical and generalized knots and provide new insights into old invariants. Much like groups arising from symmetries of geometric objects, these knot-inspired algebraic structures have connections to vector spaces, groups, Lie groups, Hopf algebras, and other mathematical structures.

In the 19th century, Lord Kelvin made the inspired guess that elements are knots in the “ether”. Hydrogen would be one kind of knot, oxygen a different kind of knot—and so forth throughout the periodic table of elements. This idea led Peter Guthrie Tait to prepare meticulous and quite beautiful tables of knots, in an effort to elucidate when two knots are truly different. From the point of view of physics, Kelvin and Tait were on the wrong track: the atomic viewpoint soon made the theory of ether obsolete. But from the mathematical viewpoint, a gold mine had been discovered: The branch of mathematics now known as “knot theory” has been burgeoning ever since.

The scientists, Gina Passante, et al., from the University of Waterloo in Ontario, Canada, have presented their experimental results for the quantum solution of the approximation of the Jones polynomial, which is a knot invariant. By approximating the Jones polynomial, researchers can determine whether two knots are different. Making this distinction is a fundamental problem in knot theory, and has applications in statistical mechanics, quantum field theory, and quantum gravity. Although approximation of the Jones polynomial is a classical problem that is very difficult to solve, the results show that the problem can be solved using a “one quantum bit model.”

As the researchers explain, the DQC1 algorithm extracts the power of one bit of quantum information alongside a register of several qubits.

“The ‘one quantum bit model’ means that there is one initialized qubit in the experiment, meaning that we can only control the initial state of one qubit,” Passante told. “For four qubits, the other three qubits are initially in a completely random state.”

The scientists experimentally implemented the algorithm using a liquid state nuclear magnetic resonance (NMR) quantum information processor. They implemented the model with the molecule transcrotonic acid, with its four carbon nuclei representing the algorithm’s four qubits. Then the researchers generated radio frequency pulses, starting randomly and improving through iterations.

“Successful experimental implementation of this algorithm relies on our ability to manipulate the qubits to perform the unitary transformations,” Passante said. “These manipulations must be done very accurately and quickly in order to get a reliable result, since the quantum states are very fragile.”

In simulations, the researchers found that, in the case of knots whose braid representations have four strands and three crossings, the algorithm could identify distinct knots 91% of the time.

Figure 2. The molecule transcrotonic acid was used in the quantum algorithm DQC1. The molecule’s four carbon nuclei represent the algorithm’s four qubits, only one of which is initially controlled. The algorithm can solve the Jones polynomial, a difficult classical problem that involves distinguishing different knots. Image copyright: G. Passante, et al. The study is published in a recent issue of Physical Review Letters. (From Phys.org.)
G. Passante, O. Moussa, C.A. Ryan, and R. Laflamme. “Experimental Approximation of the Jones Polynomial with One Quantum Bit.” Physical Review Letters 103, 250501 (2009)

As sailors have long known, many different kinds of knots are possible; in fact, the variety is infinite. A *mathematical* knot can be imagined as a knotted circle: Think of a pretzel, which is a knotted circle of dough, or a rubber band, which is the “un-knot” because it is not knotted. Mathematicians study the patterns, symmetries, and asymmetries in knots and develop methods for distinguishing when two knots are truly different.

Mathematically, one thinks of the string out of which a knot is formed as being a one-dimensional object, and the knot itself lives in three-dimensional space. Drawings of knots, like the ones done by Tait, are projections of the knot onto a two-dimensional plane. In such drawings, it is customary to draw over-and-under crossings of the string as broken and unbroken lines. If three or more strands of the knot are on top of each other at single point, we can move the strands slightly without changing the knot so that every point on the plane sits below at most two strands of the knot. A planar knot diagram is a picture of a knot, drawn in a two-dimensional plane, in which every point of the diagram represents at most two points in the knot. Planar knot diagrams have long been used in mathematics as a way to represent and study knots.

In the mid-1990s, mathematicians discovered something strange. There are Gauss codes for which it is impossible to draw planar knot diagrams but which nevertheless behave like knots in certain ways. In particular, those codes, which Nelson calls *nonplanar Gauss codes*, work perfectly well in certain formulas that are used to investigate properties of knots. Nelson writes: “A planar Gauss code always describes a [knot] in three-space; what kind of thing could a nonplanar Gauss code be describing?” As it turns out, there are “virtual knots” that have legitimate Gauss codes but do not correspond to knots in three-dimensional space. These virtual knots can be investigated by applying combinatorial techniques to knot diagrams.

Just as new horizons opened when people dared to consider what would happen if -1 had a square root—and thereby discovered complex numbers, which have since been thoroughly explored by mathematicians and have become ubiquitous in physics and engineering—mathematicians are finding that the equations they used to investigate regular knots give rise to a whole universe of “generalized knots” that have their own peculiar qualities. Although they seem esoteric at first, these generalized knots turn out to have interpretations as familiar objects in mathematics. “Moreover,” Nelson writes, “classical knot theory emerges as a special case of the new generalized knot theory.”

All colorings of the trefoil knot by the three-element kei R3 with operation table given by The matrix
. 1 2 3
_ |____
1 |1 3 2
2 |3 2 1
3 |2 1 3.

The finiteness condition for coloring objects is not required; if the target kei is infinite but has a topology, for instance, then the set of homomorphisms itself is a topological space whose topological properties then become knot invariants. Counting invariants are only the beginning; any invariant of algebraically labeled knot diagrams defines an enhancement of the counting invariant by taking the multiset of invariant values over the set of colorings of the knot.

Figure 3. Kei colorings of the trefoil by R3. Counting invariant 9 with respect to R3 and enhanced invariant p(t) = 3t +6t3. A more sophisticated example of an enhancement is the family of CJKLS quandle cocycle invariants which associate a Boltzmann weight (f ) with each quandle coloring f , determined by a cocycle in the second cohomology of T. In true combinatorial-revolutionary spirit, such two-cocycles have a geometric interpretation as virtual link diagrams.

More information: Related to this subject are an upcoming issue of the Journal of Knot Theory and its Ramifications, devoted to virtual knot theory, and the upcoming Knots in Washington conference at George Washington University, December 2-4, 2011, which will focus on on “Categorification of Knots, Algebras, and Quandles; Quantum Computing”.
Provided by American Mathematical Society
http://www.physorg.com

Figure.4. For every knot (left) an equivalent interpretation with the help silica microspheres can be found (centre). The right image shows the real experimental result. © AAAS / Science

In the theoretical part of their study they showed that for every conceivable knot a mathematically equivalent knot can be found which can be implemented in this way. "With the help of microspheres in a chiral nematic liquid crystal, we can create practically every knot that you can imagine", says Tkalec. The researchers now hope that these findings will help to better understand the complex knotting of DNA. "The knotting of DNA molecules, for example, plays an important role in many vital processes such as replication or transcription of DNA", says Uroš Tkalec. In 2006 he wrote: Colloids inducing quadrupolar order crystallize into weakly bound two-dimensional ordered structure, where the particle interaction is mediated by the sharing of localized topological defects. Colloids inducing dipolar order are strongly bound into antiferroelectric-like two-dimensional crystallites of dipolar colloidal chains. Self-assembly by topological defects could be applied to other systems with similar symmetry.

Quadrupolar order is important in DNA's.

The hadronic/mesonic strings in TGD, and how they code for the upcoming dimensions and color? Also the CP2 sheets? The scalar fundamental unity, also behind all kinds of numbers? I come back to this. Primes are open ended. But are they linked to knots too? Then this could be a hadronic string?