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Tuesday, 26.09.2017, 13.30 Uhr (WIAS-ESH)
Seminar Numerische Mathematik
Y. Ren, Dalian University of Technology, China:
On tetrahedralisations containing knotted and linked line segments
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

This talk considers a set of twisted line segments in 3d such that they form a knot (a closed curve) or a link of two closed curves. Such line segments appear on the boundary of a family of 3d indecomposable polyhedra (like the Schönhardt 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 a tetrahedralisation contains a given set of knotted or linked line segments?
In this talk, 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. When the number of twisted line segments is larger than 3, 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 twisted line segments. We show that the addition of a pair of new interior edges decomposes the original knot (or link) into two links (or knots) with less crossing numbers. 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 ≥ 3) twisted line segments. This theorem implies that the family of polyhedra generalised from the Schönhardt polyhedron by Rambau are all indecomposable.

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