| Project/Area Number |
17K05244
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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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) |
2017-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | low-dimensional topology / algebraic combinatorics / knots / directed links / directed graphs / polymatroids / polynomial invariants / quantum knot invariant / 代数的組み合わせ論 / 接触構造 / ポリマトロイド / トポロジー / 結び目理論 / 低次元トポロジー / ハイパーグラフ / 結び目不変量 / Floer homology / combinatorics |
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| Outline of Final Research Achievements |
I explored a set of ideas at the intersection of low-dimensional topology and algebraic combinatorics. On the topology side, a surprising connection between Floer theory and the Homfly polynomial was strengthened by the introduction of tight contact structures into the picture. In combinatorics, my previous theory of interior polynomials was extended in two separate directions. In the context of hypergraphs and polymatroids, the two polynomials were unified in a common two-variable extension, which is also a far-reaching generalization. The interior polynomial of an arbitrary directed graph was introduced and some of its attractive properties were discovered. Here as a special case, any undirected graph gives rise to a bidirected graph, and the interior polynomial of the latter is nothing but the h*-polynomial of the so-called symmetric edge polytope. I developed an algorithm to compute these polynomials based on an arbitrary ribbon structure of the graph.
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| Academic Significance and Societal Importance of the Research Achievements |
BernardiとPostnikovとの共同研究であるTutte多項式のまだ最も一般的なバージョンに関する結果は、文献の基本的な部分になる可能性がある。トートメレシュと共に開発した対称辺多面体のh*多項式（ひいてはh*(1)、体積）を計算するアルゴリズムは、倉本モデルとの関連から、数学以外の分野でも興味を持たれるはずである。
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