2023 Fiscal Year Research-status Report
Knot invariants and algebraic combinatorics
| Project/Area Number |
23K03108
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
KALMAN Tamas 東京工業大学, 理学院, 准教授 (00534041)
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| Project Period (FY) |
2023-04-01 – 2028-03-31
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| Keywords | Knot invariants / Algebraic combinatorics |
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| Outline of Annual Research Achievements |
Lilla Tothmeresz of Eotvos University, Hungary and I published the paper “h*-vectors of graph polytopes using activities of dissecting spanning trees” in Algebraic Combinatorics vol. 6 (2023), no. 6, pp. 1637--1651. This gives an easily computer-implementable algorithm for computing these h*-vectors or polynomials, which we have already relied on to collect data and formulate conjectures about the behavior of these objects. Our preprint "Degrees of interior polynomials and parking function enumerators" is very near publication (pending revision). In it we prove one such conjecture, on the length of the h*-vector and how it can be expressed in terms of the graph structure in many cases. We have other manuscripts in preparation. The most important one is titled "Ehrhart theory of symmetric edge polytopes via ribbon structures". This contains another approach to computing the h*-vector in perhaps its most important instance, one that is better suited to potentially prove properties such as gamma-positivity. All of these combinatorial advancements are closely related to knot invariants as well.
I have also conducted joint research with Soheil Azarpendar and Andras Juhasz of Oxford University on the trapezoidal conjecture of Fox, about the Alexander polynomial of an arbitrary alternating link, which we establish in a variety of special cases. The preprint on this work is near completion.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
All four collaborations that I described are well within my area of expertise, at the intersection of knot theory and algebraic combinatorics. One has already produced a number of publications, and in two others the writing is already underway.
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| Strategy for Future Research Activity |
Jointly with Remi Avohou of the Okinawa Institute of Science and Technology, we are working on two projects about graphs on surfaces. One aims at developing a version of Conway's polynomial for bipartite ribbon graphs, via counts of certain quasitrees of various genera. The goal of our other project is to associate a polymatroid to these same objects. We hope to achieve this by extending the notion and properties of delta-matroids to a new theory of delta-polymatroids.
I am also working on a joint preprint with Karola Meszaros of Cornell University, USA and Alexander Postnikov of the Massachusetts Institute of Technology, USA. This examines a graph-theoretical generalization of the Alexander polynomial and proves that its sequence of coefficients is log-concave.
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| Causes of Carryover |
Due to a last-minute cancellation of a trip to Korea, I did not use my entire allotment.
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Research Products
(5 results)
