The alternating virtual link database: News

Update 03/2008: links with the topology of a torus (genus 1) now have an alternate picture as a diagram drawn on a torus viewed as the doubly periodic plane. There are still some glitches.

Update 02/2006: the database is more or less back online. there might be a few glitches though.

Update 05/2004: the virtual crossings have their little circles!

Update 09/2003: finally thru with the revamping... database fully operational.

Update 06/2003: in the process of revamping the database -- one annoying feature, due to new invariants the numbering of the links might be modified. Sorry about that.

An important modification is that the normalization of the Jones polynomials is now such that P(empty)=1. In order to obtain the fairly common normalization P(unknot)=1, you must divide by the polynomial of the unknot. Jones polynomials for tangles have been similarly modified.

2-cabled Jones has now been projected onto the irreducible spin 1 subrep, which is more satisfactory. Spin 3/2 (i.e. four-dimension rep of sl(2)) Jones is also given (why? because I can!)

Remark: the reason that there are only 2 different types of spin 1/2 Jones polynomials for tangles, and not 3 as one might expect for virtual tangles (NW connecting NE, NW connecting SW, but also connecting NW connecting SE with a virtual crossing) is because the last type is forbidden by the alternating character of the tangles.

Update 05/2003: the database is ready and working.

Brief explanations

This webpage is part of the project described in the paper math-ph/0303049
``Matrix Integrals and the Generation and Counting of Virtual Tangles and Links''
by P. Zinn-Justin and J.-B. Zuber. Please refer to it for a general overview of the project.

You will find on this site a database of prime virtual alternating links (resp. tangles) up to 8 (resp. 6) crossings, with some information about them:

You can access the database by following this link. Here is a brief summary of how you can use it.

The main tool is the search. Specify various required characteristics of the links you're looking for then hit Search. (note that the "Exact Jones Polynomial" must be given in strict TeX notation with the same ordering as in the database, e.g. "-A^{-2}-A^{2}"). Click on a link number to see a particular link. (Reduced) diagrams are encoded by the permutations σ as in the paper. For compactness σ is encoded in hexadecimal. Click on diagrams for a close up (by modifying the URL you can then play around with the various options of the automated drawing program). You can also display tangles associated to a link (i.e. tangles obtained by removing one crossing from the link diagrams).

Examples: try to search for all links of genus 0 (these are "classical" links) and number of crossings less than 6. Or for all links of genus 1 with trivial Jones polynomial (max deg-min deg=1).


Back to PZJ's homepage.