• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Which presentations of (non)planar algebras give rise to knots?

+ 8 like - 0 dislike

Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is the starting point for much of quantum topology. Of course this set of generators and relations isn't unique. I'm interested in unknotting moves other than crossing changes, and I would like to ask

Is there another known "convenient" planar algebra presentation, generators modulo relations, which gives rise to knots? In particular, can I sensibly choose generators corresponding to resolutions of triple-points?

We can generalize in many ways. For example we can allow circuit algebras, which are non-planar, and obtain the set of virtual knots. I have the same question regarding such generalizations. Also

Is there a result that any presentation of a planar algebra giving rise to knots, other than the one given by crossings modulo Reidemeister moves, would necessarily be significantly harder to work with? I.e. is there some sort of non-trivial "optimality result" for the presentation "crossings mod Reidemeister moves"?

This post imported from StackExchange MathOverflow at 2014-09-21 14:26 (UCT), posted by SE-user Daniel Moskovich

asked Aug 17, 2010 in Theoretical Physics by Daniel Moskovich (130 points) [ revision history ]
edited Sep 21, 2014 by Dilaton
You should look at Example 2.5 of Vaughan Jones's "Planar Algebras." There he works out a presentation of the HOMFLY skein theory starting with generators which are resolutions of triple points. Remark 1 there seems to be essentially your question, so I don't think such a presentation is known.

This post imported from StackExchange MathOverflow at 2014-09-21 14:26 (UCT), posted by SE-user Noah Snyder
Sorry, forgot the link: math.berkeley.edu/~vfr/planar.pdf

This post imported from StackExchange MathOverflow at 2014-09-21 14:26 (UCT), posted by SE-user Noah Snyder
Noah: Thanks for the link! I knew nothing about this. Yes, his Remark 1 does turn out to be my question, with similar motivation..

This post imported from StackExchange MathOverflow at 2014-09-21 14:26 (UCT), posted by SE-user Daniel Moskovich

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights