Research on spatial graphs in the simple cubic lattice and its application to polymer science

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K21827
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 11:Algebra, geometry, and related fields
• Research Institution: Ochanomizu University (2022); Saitama University (2019-2021)
• Principal Investigator: Shimokawa Koya お茶の水女子大学, 基幹研究院, 教授 (60312633)
• Project Period (FY): 2019-06-28 – 2023-03-31
• Keywords: 空間グラフ / 格子モデル / 多環状高分子

Outline of Final Research Achievements

We classified BFACF moves for lattice spatial graphs whose vertices have degree 3. We also proved that for lattice space graphs, the equivalence class by BFACF moves and the equivalence class by ambient isotopy of the space graphs coincide. This indicates that by properly defining BFACF moves, the lattice knot and entanglement results were extended to lattice space graphs.
Since the BFACF move for spatial graphs is a planar move and the BFACF move can be considered for graphs in the 2-D plane as well, we first simulated the case of graphs in the 2-D plane.

Free Research Field

トポロジーとその応用

Academic Significance and Societal Importance of the Research Achievements

この研究では、近年合成されている複雑な構造をもつ高分子のトポロジーの構造の数学的モデルを扱っている。今回の成果は、トポロジーの一分野である結び目理論の研究を行ったもので、立方格子内の空間グラフのトポロジーに関するものである。応用として多環状高分子、タンパク質の立体構造、DNAのR-ループへの応用が見込めるものとなっており、今後様々な分野にわたる発展が期待される。