Knot theory Definition and 19 Threads

I am highly interested in Topological Quantum Field Theory (TQFT) and am currently planning on doing a project on this topic this year. Some of my relevant background: Algebra (Groups, Rings, Fields, basics of Categories and Modules), Topology (Munkres), Smooth Manifolds (John Lee's book, first...

The PD code [(2, 3, 1, 4), (4, 1, 3, 2)] seems to map to a non-unique knot diagram. I can describe the following two Hopf links with different orientations with this same PD code. As I understand it, while a link diagram does not have a unique PD code, a given PD code should map to just one knot...

The standard configuration of Brunnian "rubberband" loops shows a series of unknots each bent into a U-shape, with their ends looped around the middle of the next unknot. (See for instance http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links). This connection requires 8...

Hello! Where can I find the source for this statement (i.e. a citation for it, ideally the original one): "any smooth k-sphere embedded in ##R^n## with 2n − 3k − 3 > 0 is unknotted". Thank you!

I'm not a math machine, but I dabble in dimensional stuff. I think this falls under knot theory. I have built several prototypes of a tesseract. Each of them sits in a little case in my office. One of them is made from truncated cubes, held together with elastic cord: In theory, the...

Advanced Physics (Advanced Science) by Steve Adams & Jonathan Allday from OUP Oxford: and Physics (Collins Advanced Science) 3rd Edition by Kenneth Dobson from Collins Educational: http://www.amazon.com/dp/0007267495/?tag=pfamazon01-20 Does anyone know any of these books? I find them very...

I refer to the list here: https://math.berkeley.edu/~kirby/ entitled "Problems in Low-Dimensional Topology" Question says it all. I am giving a graduate level presentation to a group that includes some knot theorists. (I believe the collective term for knot theorists is a "tangle"...

I have looked in vain on the web for pictures of magnetic field lines for multiple linked current loops. I would be happy just to see a picture of the field lines for a simple Hopf link but somewhere there must be pictures for the Borromean rings and other more complex links - and also braids...

The title above give my name. I am a pure maths PhD with an interest in physics and geometry. I am currently studying physics for fun and I am very interested in current progress. I am especially interested in quantisation of space time, holographic theories and dualities. Regards John

How is knot theory involved in other fields of math? Any applications outside of math?

I want one good for physicists does not need to be very rigorous

Probably a bit abstract,but I was thinking if 4D closed strings could form knots? I mean if a closed string in 4-dimensional spacetime can be considered an unknot and a knot polynomial be associated with every closed string. I also wondered that if the fundamental strings vibrate in the knotted...

I need a free description with illustrations on 4D knots theory, especially the 4D generalization of Reidermeister moves and the movie represantation. Where can I find a freely available paper?

I'm looking for an introductory book for knot theory. I have background in topology and algebraic topology. I would prefer a more sophisticated treatment, but I have no previous knowledge.

Hi, I was thinking about Knot Theory for a while and started thinking about higher dimensionalities. Could the knots we know so well (knots in 3d space) be undone if allowed to be manipulated through a fourth spatial dimension? Could they be made topologically equivalent to the unknot? And if...

So I know that quandles are associated with knot theory, etc. but what are "shadow colorings" ? on a broader context, can someone please give me a simple definition of a "knot invariant" and a "quandle" ? and what does this have to do with "racks" ? sorry, all this esoteric language is...

i cannot find a proof anywhere to show that the linking number is always an integer! can someone please point me in the right direction (or give a proof of their own)! thanks.

Hi, so I need to show that every link is two-equivalent to a trivial link with the same number of components. Right now I can show that if I have a simple link with linking number = 1, then it is possible to immediately separate the link into its two components. But how can I generalize this...

Hi, I'll attend a lecture around 14 hours later addressed by a Nobel laureate and here's the brief description of the lecture. "Starting from a Parlor Game, I shall show how a deep mathematical problem can be formulated in an elementary way. The steps are understandable to high school...