Talk:Linking number

Major Revision

I have revised the article thoroughly, adding several pictures and incorporating material from the old linking coefficient article. The material on self-linking number has been moved to a separate article. Jim 03:05, 26 July 2007 (UTC)

This article doesn't seem to include the linking of two simple closed curves in 3-manifolds other than the 3-sphere. In general, such linking numbers will be fractions, or simply undefined, depending on the homology of the curves. -- Ken Perko 69.113.192.109 (talk) 04:00, 24 July 2015 (UTC) (lbrtpl@gmail.com)

A good explanation of linking numbers in 3-manifolds is contained in Section 77 of Seifert and Threlfall's "Textbook of Topology" (available for free on the internet). They exist whenever both curves are rationally null-homologous and are calculable by constructing a surface bounded by one and counting its intersections with the other. -- Ken Perko — Preceding unsigned comment added by 24.44.60.33 (talk) 02:46, 16 April 2019 (UTC)

Linking numbers between branch curves of non-cyclic covering spaces of knots, introduced by Reidemeister, were an important invariant for classifying knots with more than eight crossings (the homology of cyclic covers, considered by Alexander and Briggs, having failed to suffice at this level of complexity). They seem to do the trick for all knots looked at thus far.

== The gauss formula for the linking number (second line) appears to have some typo. The argument of the determinant has three elements? It should be dgamma1/dt x dgamma2/dt dot (gamma1 - gamma2) or something like that! — Preceding unsigned comment added by 86.14.201.34 (talk) 15:45, 20 April 2023 (UTC)

See also

Kissing number? How the hell is that supposed to be related? —Preceding unsigned comment added by 141.84.69.20 (talk) 22:27, 11 December 2009 (UTC)

Since when are cup products a generalization of linking numbers???

One sentence under Generalizations reads as follows:

"In algebraic topology, the cup product is a far-reaching algebraic generalization of the linking number, with the Massey products being the algebraic analogs for the Milnor invariants."

To the best of my knowledge, cup products (of cohomology classes) in manifolds are not directly relevant to any linking numbers. Rather, they are dual to the intersection pairing on homology.

(It is true that the intersection of homology classes can be used in the definition of linking number. But the intersection of homology classes is definitely not the same as linking number per se.

So, this sentence needs to be radically altered.Daqu (talk) 04:32, 30 January 2013 (UTC)

Could I add a small snippet about how the Linking number relates to DNA?

Hello there, my name is Martin Lubell, and I'm starting with my Masters program in Bioinformatics at Johns Hopkins University. In our Molecular Biology of the Gene class, we are also beginning to learn how to edit Wikipedia articles. I was wondering if you would allow me to add a section on how the Linking Number relates to DNA. Thank you, m a r t i n MartinLubell (talk) 06:08, 13 February 2013 (UTC)

Hi Martin. You are welcome to add a section on anything that is relevant to the page. Moreover, you don't need to ask permission - Wikipedia encourages new editors to be bold! Anything you write can be reverted if the worst comes to the worst, though I'm sure that won't be necessary! 2001:630:E4:42F9:1:5643:9330:D77E (talk) 17:37, 25 February 2013 (UTC)

The sign of the linking number in the integral form does not coincide with the original definition by Gauss

As it stands at this moment the linking number in the integral form does not coincide with the original definition by Gauss. Indeed the natural orientation of the unit sphere is given by the outwards normal vector, while the orientation of the torus is given by the order of circles in the product of two circles. In such settings the Gauss linking integral will be negative to the one written in the article.

All the best, Oleg Karpenkov

Original Gauss definition is from (p.605): Zur mathematischen Theorie der electrodynamischen Wirkungen, manuscript, first published in his Werke Vol. 5, JOURNAL = {K\"onigl. Ges. Wiss. G\"ottingen, G\"ottingen}, YEAR = {1877}, NUMBER = {1}, PAGES = {601--629} 149.241.92.220 (talk) 18:59, 5 October 2023 (UTC)

Generalization of cup product?

The section Generalizations contains this sentence:

"In algebraic topology, the cup product is a far-reaching algebraic generalization of the linking number, with the Massey products being the algebraic analogs for the Milnor invariants."

But the linked article Cup product does not explain the connection between cup product and linking number. (Although it gives one vague example.)

I hope that someone knowledgeable about this subject can fix this omission.