Development of tangle-method in surface-knot theory and its applications

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400090
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kobe University
• Principal Investigator: Satoh Shin 神戸大学, 理学(系)研究科(研究院), 教授 (90345009)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 曲面タングル / 曲面結び目 / 溶接結び目 / フォックス彩色 / デーン彩色

Outline of Final Research Achievements

A surface-knot is a closed surface embedded in 4-space. We study many properties of surface-knots from the viewpoint of surface-tangles and local moves. We characterize a surface-knot which has a decomposition of trivial surface-tangles, and prove the splittability of a surface-tangle with fabric structure.
On the other hand, we also study many properties of virtual knots and welded knots which present surface-knots of genus one. We characterize a virtual knot whose 3-state number is zero, and determine the minimal number of colors needed for effective Fox or Dehn colorings for a knot or surface-knot. We also prove that the crossing change and delta-move on a welded knot are both an unknotting operation.

Free Research Field

結び目理論