Science topics: Geometry and TopologyKnot Theory

Science topic

# Knot Theory - Science topic

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3.

Questions related to Knot Theory

Does anyone know of applications of multiple sums of a sequence?

I know of the Multiple Zeta values (which is a multiple sum of 1/N^s). This has multiple applications in quantum physics, QED, QCD, connection between knot theory and quantum physics, ...

Does anyone know of potential applications for this more general form which is a general multiple sums? I have written an article about it and about its applications including partition identities, polynomial identities. I wanted to know if anyone know of applications outside mathematics or additional applications in math.

A generalized pretzel knot (or link) is an unoriented link that, up to isotopy, bounds an unoriented, possibly non-orientable surface P(t_1,...,t_k) formed by joining two parallel 2-disks in horizontal planes in R^3 with k bands that have vertical line segments as their core arcs, where the ith band has t_i half-twists. (A classical pretzel has k = 3; if k = 2 you get a torus knot or link of type (2,t_1+t_2); if k = 1 you get an unknot.) I would like an algorithm (if possible) to pass from (t_1,...,t_k) to a closed braid representation of any one of (or all of?) the oriented links that can be obtained by variously orienting the components of the generalized pretzel P(t_1,...,t_k).

An interesting, possibly suggestive, example is P(-3,3,-2). It is a non-orientable surface. Its boundary is the knot 8_20, a ribbon knot. The ribbon disk it bounds corresponds to the "band representation" (my language) (b(1),b(2)) in the 3-string braid group B_3, where b(1)=\sigma_1 and b(2)=(\sigma_2)^3\sigma_1(\sigma_2)^{-3}; in particular, the closure of the braid

\sigma_1(\sigma_2)^3\sigma_1(\sigma_2)^{-3} is 8_20. Furthermore, if you draw the standard closed-braid diagram of that length 8 3-string braid, its non-orientable checkerboard surface is isotopic (in the plane, or on S^2, depending on how you close your braids) to P(-3,3,-2). So in this one case (and of course a passel of very similar ones) there's an algorithm, sort of. But I don't see any way to make it much more general.

Google found me an article by a Polish physicist called, hopefully, "Braids for Pretzel Knots", but it doesn't seem helpful to me (and the most promising among his references don't actually appear to have the content that say what he says they say, although since I don't speak Physics the problem may be mine, not his). Other than that, Google found nothing. So here I am.

I found local move

identities of polynomial invariants for high dimensional knots.

I found

Alexsander(K_+)-Alexsander(K_-)=(t-1)Alexsander(K_0)

for a local move and high dimmensional knots K_+, K_-,K_0,

which is indicated in my paper below.

Local move identities for the Alexander polynomials of high dimensional knots and inertia groups

Journal of knot theory and its ramificatioms vol18, no.4 (2009) 531-545, math.GT/0512168, UTMS 97-63

In this paper I also found

Arf(K_+)-Arf(K_-)=(|bP_{4k+2} ∩ I(K_0)| + 1) mod 2,

for a local move and high dimmensional knots K_+, K_-,K_0,

where bP is the bP-subgroup of homotopy sphere groups and

I is the innertia group.

Furthermore we have

Alexsander(K_+)-Alexsander(K_-)=(t+1)Alexsander(K_0)

for a local move, which is indicated below.

Note the right hand side. It is (t+1) not (t-1). It is a new type.

Local moves on knots and products of knots (with Louis H. Kauffman)

Knots in Poland III-Part III Banach Center Publications Volume103 (2014), 159-209 Institute of Mathematics Warszawa 2014 arXiv:1210.4667[math.GT]

Can String theory and QFT make an interpretation of the above local move

identities of polynomial invariants for high dimensional knots as in the 1-link case?

I would like to know if there are any examples where the HOMFLY polynomial fails. Like two knots having the same polynomial when they shouldn't, or having different ones when they should be the same (this one is not possible I think, since then HOMFLY shouldn't be called invariant any more), or maybe chirality issues. Both Alexander and Jones have their breakpoints, so I assume HOMFLY does too.

What I've found until now is that we can not prove that HOMFLY polynomial is not perfect, but it is not likely to be perfect. I just wonder if anyone found a point where it fails, that I couldn't, an example that proves it wrong.

Is the projection of any spherical 2-knot that of a trivial 2-knot?

It is very well-known that the projection of any 1-knot is that of a trivial 1-knot.

I proved that for an integer n>2, there is a spherical n-knot whose projection is not the projection of any trivial knot. See the following paper.

Furthermore I proved that for an integer n>4, there is a spherical n-knot with the following properties: the projection is not the projection of any trivial knot. The singular point set consists of double points. The number of connected components of the singular point set is two. See the following paper.

Another question is: How about the n=3,4 case of this result?

The projections of n-knots which are not the projection of any unknotted knot

*Journal of knot theory and its ramificatioms 10 (2001) 121-132*, math.GT/0003088, UTMS 97-34. Singularities of projections of n-dimensional knots

*Mathematical Proceedings of the Cambridge Philosophical Society,*126 (1999) 511-519, UTMS96-39, arXiv:1803.03221We produce pulp from cotton and now, there is a problem for separation remain cotton knots from obtained pulp. What are possible solutions to solve this problem, Deflaker,Centrifugal Cleaner, Turbo separator,..?

How we can determine the control points and knot vectors required to represent the geometry of a structural member such as a square plate, a circular plate etc .

I am trying to understand more about knots in higher dimensions. I understand that in general one wants co-dimension 2 so that, for example, S^2 can be knotted in S^4.

What I don't understand is to what extent the notions and theorems relating to knot complements and knot groups carry over to higher dimensions. Specifically I would like to know if the Gordon-Luecke theorem which states that "a knot is determined by its complement" is valid in general or only for S^1 knots in S^3.

I have also come across (twisted) spun knots and am wondering if there exists a one-to-one mapping between ordinary S^1 knots and spun knots. Presumably one could have higher dimensional knots that are not spun knots....?

Any insight would be greatly appreciated!

See the enclosed paper by Jack Avrin, where he discusses his knot model of elementary particles. Since Knot theory corresponds to fluid dynamics, and fluid dynamics theory can be shown to correspond to electromagnetic theory, then his model seems plausible at least.

Is it possible to come up with a knot model of elementary particles? What do you think? Your comments are welcome.

Quandles and racks are algebraic structures referred to in knot theory. Are there links to thorough introductions for researchers in how they are used in physics research?

Knot theory is an interesting area of mathematics studying the unique properties of topological morphs in knot attribute. It is an investigation into the way to extend the one dimensional string to the wider implications in the higher dimensional space. The attached book provides an excellent reference:

His work on knot polynomials, with the discovery of what is now called the Jones polynomial, was from an unexpected direction with origins in the theory of von Neumann algebras. But how is his work related to statistical mechanics?

I researched and found some articles regarding symbolic dynamics and periodic orbits, let me know if there is a close relationship between these disciplines.