
On Tetrahedralisations Containing Knotted and Linked Line Segments
Si, Hang, Yuxue Ren, Na Lei, Xianfeng Gu
Proceedings, 26th International Meshing Roundtable, Elsevier, Science Direct, September 1821 2017

INTERNATIONAL MESHING ROUNTABLE

26th International Meshing Roundtable
Barcelona, Spain
September 1821, 2017
Hang Si, WIAS Berlin, DE, Hang.Si@wiasberlin.de
Yuxue Ren, School of Mathematics, Jilin University, CN, snow_ren@foxmail.com
Na Lei, School of Software Technology, Dalian University of Technology, CN, nalei@dlut.edu.cn
Xianfeng Gu, Computer Science Department, Stony Brook University, US, gu@cs.stonybrook.edu
Abstract
This paper considers a set of twisted line segments in 3d such that they can form a knot (a circle) or a link of two circles. Such line segments appear on the boundary of a family of 3d {\it indecomposable} polyhedra (like the Schonhardt polyhedron) whose interior cannot be tetrahedralised without additional vertices added. On the other hand, a 3d (nonconvex) polyhedron whose boundary contains such line segments may still be decomposable as long as the twist is not too large. It is therefore interesting to consider the question: when there exists or does not exist a tetrahedralisation contains a given set of knotted or linked line segments? In this paper, we studied a simplified question with the assumption that all vertices of the line segments are in convex position. It is straightforward to show that no tetrahedralisation of $6$ vertices (the threelinesegments case) can contain a trefoil knot. Things become interesting when the number of line segments increases. Since it is necessary to create new interior edges to form a tetrahedralisation. We provided a detailed analysis for the case of a set of $4$ line segments. This leads to a crucial condition on the orientation of pairs of new interior edges which determines whether this set is decomposable or not. We then prove a new theorem about the decomposability for a set of $n$ ($n \ge 3$) knotted or linked line segments. This theorem implies that the family of polyhedra generalised from the Schonhardt polyhedron by Rambau~\cite{Rambau05} are all indecomposable.
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