# 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

Hang Si, WIAS Berlin, DE, Hang.Si@wias-berlin.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 (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.