Department of Computer Science and Engineering On Computing handles and tunnels for surfaces Tamal K. DeyKuiyu LiJian Sun

2/10 Department of Computer Science and Engineering Definition Notation: M: a connected compact surface in R 3. I: the bounded set enclosed by M including M; O: the unbounded set. tunnel loop: trivial in H 1 (O) and non-trivial in H 1 (I). handle loop: trivial in H 1 (I) and non-trivial in H 1 (O). i * : H 1 (M)  H 1 (O) and i * : H 1 (M)  H 1 (I)

3/10 Department of Computer Science and Engineering Surface often used to represent the solid it enclosed useful to study the loops on M which are non trivial in I or O Usefulness of the concepts of handle and tunnel loops Nontrivial loops

4/10 Department of Computer Science and Engineering Loops on a surface Compute polygon schemas [Vegter-Yap90][Dey-Schipper95] Linear time algorithm Compute optimal systems of loops [VL05][EW06] Verdiere and Lazarus gave an algorithm for computing a system of loops which is shortest among the homotopy class of a given one. Erickson and Whittlesey gave a greedy algorithm to compute the shortest system of loops among all systems of loops. Compute optimal cut graph [EH04] Computing a shortest cut graph is NP-hard. Related work

5/10 Department of Computer Science and Engineering System of handle loops: a set of loops {h i } i=1 n on M that are trivial in H 1 (I) but form a basis for H 1 (O). System of tunnel loops: a set of loops {t i } i=1 m on M that are trivial in H 1 (O) but form a basis for H 1 (I). Existence H 2 ( R 3 )  H 1 (M)  H 1 (I) © H 1 (O)  H 1 ( R ) 0  H 1 (M)  H 1 (I) © H 1 (O)  0 H 1 (M) ' H 1 (I) © H 1 (O) which is induced by inclusion. [Moise77] rank(H 1 (I)) ¸ g and rank(H 1 (O)) ¸ g, which implies n=m=g. Existence

6/10 Department of Computer Science and Engineering Consider a thickened trefoil handle loop (green) tunnel loop Knotted loops

7/10 Department of Computer Science and Engineering Graph retractability assumption A surface is graph retractable if both I and O deformation retract to a graph, denoted I and O, respectively.

8/10 Department of Computer Science and Engineering I and O are two disjoint graph, each of which contains g number of loops (g: the genus of M). Denote these loops by {K j I } j=1 g and {K j O } j=1 g or simply {K j } j=1 2g. How to compute I and O ? Inner and outer graphs

9/10 Department of Computer Science and Engineering Linking number J and K are two disjoint knots. L be the link formed by J and K. In a regular projection of L, there are two ways in which J crosses under K. The linking number, lk(K, J), is the sum of these signed crossings.

10/10 Department of Computer Science and Engineering Computable criterion Theorem 1: A loop  on M is a handle one iff lk( , K i I )  0 for at least 1 · i · g and lk( , K i O )=0 for all 1 · i · g. A loop  on M is a tunnel one iff lk( , K i O )  0 for at least 1 · i · g and lk( , K i I )=0 for all 1 · i · g.

11/10 Department of Computer Science and Engineering Existence Theorem 2: There exist 2g loops, denoted {J j } j=1 2g, such that lk(K i, J j )=  ij. Half of them linked with K i I ’s, denoted {J j I } j=1 g, are handle loops, and the other half linked with K i O ’s, denoted {J j O } j=1 g, are tunnel loops. {[J j I ]} j=1 g form a basis for H 1 (I) and {[J j O ]} j=1 g form a basis for H 1 (O). Hence {[J j ]} j=1 2g form a basis for H 1 (M).

12/10 Department of Computer Science and Engineering Topological algorithm Assume I and O are given. Step1: Compute {K i } i=1 2g using the spanning tree of I and O Step2: Compute a system of 2g loops on M, denoted {  j } j=1 2g. Step3: Compute lk(K i,  j ) for all i and j. Let A be the matrix {lk(K i,  j )}. A is the transform matrix from basis {[J j ]} j=1 2g (defined in Theorem 2) to basis {[  j ]} j=1 2g. A -1 = {a ji } exists and has integer entries. [J j ] =  i=1 2g a ji [  i ]. Step4: Obtain J j by concatenating  i ’s according to the above expression.

13/10 Department of Computer Science and Engineering An implementation to compute system of handle and tunnel loops with small size Compute I and O Basic idea: Collapse the inside (outside) Voronoi diagram to obtain I ( O ). The curve-skeleton [DS06] captures the geometry better. Each skeleton edge (e) gets associated with an addition value called geodesic size (g(e)) indicating the local size of M. Establish graph structure on the curve skeleton. The geodesic size for a graph edge, g(E) = min{g(e): e is a skeleton edge in E}. Issue: not always work for any graph retractable surface, e.g., a thickening of a house with two room.

14/10 Department of Computer Science and Engineering An implementation to compute system of handle and tunnel loops with small size Compute {K j } j=1 2g Compute the maximal spanning tree for I and O using geodesic sizes as weight. Add the remaining edges, E i ’s, to form K i ’s. Compute {J j } j=1 2g Basic idea: compute J i ’s at different location indicated by E i ’s. Let e be the skeleton edge with the smallest geodesic size in E i. Let p be one of the vertices of the dual Delaunay triangle of e. Compute an optimal system of loops [EW06] and apply topological algorithm. In all our experiments, one of the loop in the system of loops itself satisfies the condition to be J i ’s.

15/10 Department of Computer Science and Engineering Results

16/10 Department of Computer Science and Engineering Conclusion and future work