Yes, because the chord diagram algebra is the Pontrjagin ring of the loop group: The composition in the loop group is only $A_\infty$-associative, but under passage to homology this becomes an associative algebra.

]]>In 1912.10425, intuitively what explains the fact that the physically-meaningful object is the *associative* algebra of chord diagrams? More precisely, how can we see we can just take the concatenation to be associative, unlike virtually any discussion related to say concatenation of paths (where one has to quotient by homotopy)? Is it because this describes the *topological* phase space?

there is an evident general encoding of hyperbolic tesselations by round chord diagrams.

Is this made explicit anywhere?

In particular, I am wondering about the following:

It should be possible to obtain the round chord diagram for an $\{n,k\}$ tesselation at some finite stage by starting with a suitable horizontal chord diagram on $n-1$ strands, then iteratively “cabling” it into itself and forming vertical composites of that with itself, and finally taking the trace to a round chord diagram with a cyclic permutation.

Is this described anywhere?

]]>added pointer to these two, which might be close to the earliest appearance of the concept of chord diagrams (?):

Paul R. Stein,

*On a class of linked diagrams, I. Enumeration*, Elsevier Journal of Combinatorial Theory, Series A Volume 24, Issue 3, May 1978, Pages 357-366 (doi:10.1016/0097-3165(78)90065-1)Paul R. Stein, C. J. Everett,

*On a class of linked diagrams II. asymptotics*, Discrete Mathematics Volume 21, Issue 3, 1978, Pages 309-318 (doi:10.1016/0012-365X(78)90162-0)

added pointer to today’s

- Ali Assem Mahmoud, Karen Yeats,
*Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions*(arXiv:2010.06550)

added pointer to today’s

- Noboru Ito,
*Based chord diagrams of spherical curves*(arXiv:2004.12121)

made explicit the definition of $\mathcal{A}^c$ (here)

]]>I have replaced the graphics illustrating a typical chord diagram (first graphics in the entry) with a new one (here) that may be a little more informative

]]>In a new Properties section *Relation to Chern-Simns diagrams* I added pointer to *chord diagrams modulo 4T are Chern-Simons diagrams modulo STU*.

No other text yet. To be expanded. But not tonight.

]]>added these pointers:

Discussion of chord diagrams encoding SYK model correlators as representing hyperbolic/holographic content:

{#BINT18} Micha Berkooz, Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of:

*Towards a full solution of the large $N$ double-scaled SYK model*, JHEP 03 (2019) 079 (arxiv:1811.02584){#Narovlansky19} Vladimir Narovlansky, Slide 23 (of 28) of:

*Towards a Solution of Large $N$ Double-Scaled SYK*, Nazareth 2019 (pdf, NarovlanskySYK19.pdf:file)

following

- Micha Berkooz, Prithvi Narayan, Joan Simón,
*Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction*, JHEP 08 (2018) 192 (arxiv:1806.04380)

Fixed the diagrams now, hopefully. [Edit: there is something strange going on which causes the diagrams not to render correctly after an edit, but to render correctly after I re-render manually. I will look into it when I get the chance, probably this evening.]

]]>I created a short page for chord diagram, and also added a bit of relevant information to Vassiliev invariant.

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