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\par\noindent{\Large\bf Emergent Knots in a Pole Dancing Studio \hfill SMRI Visit Proposal for 2025}
\vskip 2mm

\begin{multicols}{2}

It's a bit tough to write a research proposal for research that will take place more than a year from now. In the optimistic scenario, we will finish everything that's on our desks right now, and be ready to sail on into the horizon. In the pessimistic scenario, we will still be struggling with the details of a paper we started 10 years ago. In the most realistic scenario, we will be pushing our understanding of {\bf emergent knots} (and other knotted objects: links, braids, tangles, etc.) in a {\bf pole dancing studio}.

\parpic[r]{\input{PDS3.pdf_t}}
What's a {\bf pole dancing studio}? It's a big room (namely, a disk cross an interval) with a few stationary poles (namely, straight vertical lines going from floor to ceiling) removed. A picture is on the right, and in dryer language, it is simply a punctured disk cross an interval. Let's call the pole dancing studio with $n$ poles $\PDS_n$.

What's an {\bf emergent knot}? It's an object living in a quotient of knot theory in which almost all knottedness is killed, and only a tiny bit remains. Precisely, for the strands of the knotted object itself (not for the poles), we mod out by the relation $\doublepoint\doublepoint=0$, where $\doublepoint\coloneqq\overcrossing-\undercrossing$ (namely, a knotted object with two double points is set to $0$, where a double point is a mnemonic for the difference between an over crossing and an under crossing). Had we been moding out by the stronger relation $\doublepoint=0$, we would be declaring that an over crossing is equal to an under crossing, and that kills all knottedness. We are not that cruel, and we let a whiff of knottedness survive. Let us call the space of emergent knots $\calK_1$.

Collapsed knots, namely knots modulo $\doublepoint=0$, are simply homotopy classes of curves in $\PDS_n$. If we project them to the floor of the studio we lose nothing and we see curves in a punctured disk, namely, elements of the free group $\FG_n$ (at this level of detail we are ignoring base points). The free group has a commutative algebra analog, the free associative algebra $\FA_n$, and there is the so-called Magnus expansion $Z_0\colon\FG_n\to\FA_n$ which plays a significant role in combinatorial group theory.

It turns out that the procedure that takes $\FG$ to $\FA$ (``associated graded'') can be imitated in the case of emergent knots, and then it takes $\calK_1$ to $\calA_1$, ``emergent chord diagrams'', which we will not define here, except to say that it is a whiff more than the free associative algebra $\FA$. And it makes sense to ask the question, is there an expansion $Z_1\colon\calK_1\to\calA_1$? Is there such an expansion which is homomorphic, meaning, which preserves several additional structures that $\calK_1$ and $\calA_1$ inherit from knots?

The answer is YES. Furthermore, it turns out that $Z_1$ allows us to construct an expansion for the Goldman-Turaev Lie bialgebra, a certain algebraic structure constructed from curves in the punctured disk. In itself, this was shown by Alekseev, Kawazumi, Kuno, and Naef to be equivalent to a solution of the Kashiwara-Vergne problem which in itself implies an equivalence statement between convolutions of invariant functions on Lie groups and on Lie algebras. We don't know a direct relation between emergent knots and this convolutions statement. We will certainly try to find this relationship during my proposed visit to SMRI!

By an earlier work of Dancso and myself, that same Kashiwara-Vergne problem is also related to ``w-knots'', a class of 2-dimensional knotted objects living in 4-dimensional space. Thus emergent knots ought to be related to w-knots, but we don't know a direct relation. Please add that to our SMRI to do list.

It is well known that expansions for (``full'', not merely emergent) knots are closely related to Drinfel'd associators and to the Grothendieck-Teichm\"uller group $GT$. There are analogs of these notions for emergent knots, and seeing that in emergent knots almost all complexity is killed off, emergent associators are simpler to compute than full associators and the study of $GT$ also becomes simpler. Yet recent computations by Kuno and myself show that up to degree 10 or so, the simpler emergent theory is equal to the harder full theory. We don't know why this is so. Perhaps we will figure it out in 2025!

\end{multicols}

Thus I hope that SMRI will extend its hospitality to me and to my research with Dr.\ Zsuzsanna Dancso, and will fund my visit to Sydney between May 24 2025 and July 6 2025. During that time I will be delighted to speak about this subject in Sydney. I expect that I will spend most of that period in Sydney, though it is possible that I will visit Melbourne and/or Brisbane for 2-3 days each.

A handout from a talk I gave on the subject in Switzerland in 2022 is on the following two pages. It has many more details, and a few relevant references. A video of that talk is linked at \url{http://drorbn.net/ld22}.

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