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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 18-21 2017


26th International Meshing Roundtable
Barcelona, Spain
September 18-21, 2017

Hang Si, WIAS Berlin, DE,
Yuxue Ren, School of Mathematics, Jilin University, CN,
Na Lei, School of Software Technology, Dalian University of Technology, CN,
Xianfeng Gu, Computer Science Department, Stony Brook University, US,

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 (non-convex) 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 three-line-segments 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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