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Dear topologists, </div>
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A few people responded positively, so Mathijs is going to give a talk on </div>
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<div dir="ltr" role="presentation">Wednesday 17th December</div>
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<span style="font-family: "helvetica neue"; font-size: 13px; color: rgb(33, 33, 33);"><b>Title:
</b>Braiding vineyards</span></p>
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<span style="font-family: "helvetica neue"; font-size: 13px; color: rgb(33, 33, 33);"><b>Abstract:</b> In this work, we introduce and study what we believe is an intriguing and, to the best of our knowledge, previously unknown connection between two fundamental
areas in computational topology, namely topological data analysis (TDA) and knot theory. Given a function from a topological space to R, TDA provides tools to simplify and study the importance topological features: in particular, the lth-dimensional persistence
diagram encodes the topological changes (or l-homology) in the sublevel set as the function value increases into a set of points in the plane. Given a continuous one parameter family of such functions, we can combine the persistence diagrams into an object
known as a vineyard, which track the evolution of points in the persistence diagram as the function changes. If we further restrict that family of functions to be periodic, we identify the two ends of the vineyard, yielding a closed vineyard. This allows the
study of monodromy, which in this context means that following the family of functions for a period permutes the set of points in a non-trivial way. Recent work has studied monodromy in the directional persistent homology transform, demonstrating some interesting
connections between an input shape and monodromy in the persistent homology transform for 0-dimensional homology embedded in R^2. </span></p>
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<span style="font-family: "helvetica neue"; font-size: 13px; color: rgb(33, 33, 33);">In this work, given a link and a value l, we construct a topological space (based on the given link) and periodic family of functions on this space (based on the Euclidean
distance function), such that the closed l-vineyard contains this link. This shows that vineyards are topologically as rich as one could possibly hope, suggesting many future directions of work. Importantly, it has at least two immediate consequences we explicitly
point out:</span></p>
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Monodromy of any periodicity can occur in a l-vineyard for any l. This answers a variant of a question by Arya and collaborators. To exhibit this as a consequence of our first main result we also reformulate monodromy in a more geometric way, which may be of
interest in itself.</li><li style="font-family: "helvetica neue"; font-size: 13px; color: rgb(33, 33, 33); line-height: normal; margin: 0px;">
Topologically distinguishing closed vineyards is likely to be difficult (from a complexity theoretical as well as practical perspective) because of the difficulty of knot and link recognition, which have strong connections to many NP-hard problems.</li></ul>
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For those interested, there will be time for discussion afterwards. </div>
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Best,</div>
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Senja</div>
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<b>Van: </b>Barthel, S.D. (Senja) <s.barthel@vu.nl><br>
<b>Datum: </b>donderdag, 20 november 2025 om 16:11<br>
<b>Aan: </b>_list_topology.beta <topology.beta@listserver.vu.nl><br>
<b>Onderwerp: </b>Visit Mathijs Wintraecken - braiding vineyards<br>
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Dear all, </div>
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<div dir="ltr" class="ms-outlook-mobile-reference-message skipProofing"><span style="font-family: Aptos, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">On 17th December Mathijs Wintraecken is visiting me to discuss possible relations between
</span><span style="font-family: Aptos; font-size: 16px; color: rgb(33, 33, 33);">knotting and braiding in vineyards and DNA topology.</span><span style="font-family: Aptos, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
<a href="https://www-sop.inria.fr/members/Mathijs.Wintraecken/" data-outlook-id="e9c44f50-a93b-4896-a47e-efb0bf58c5cb">
https://www-sop.inria.fr/members/Mathijs.Wintraecken/</a> </span></div>
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He recently finished a paper on braiding vineyards <span style="color: rgb(0, 120, 215);">
<a href="https://arxiv.org/pdf/2504.11203" data-outlook-id="c6a17248-0ff8-40d4-81ef-3c4b7bf372a8" style="color: rgb(0, 120, 215);">https://arxiv.org/pdf/2504.11203</a></span></div>
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and has a presentation on the topic.</div>
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Please let me know if you are around on this day and interested in attending such a talk, and also if you would want to meet him\join the discussion. </div>
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Best,</div>
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Senja</div>
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