arXiv subjects

James Cruickshank

Publications and source records attributed to James Cruickshank.

21 records · Page 2Linked to original sources

arXiv2015

The generic rigidity of triangulated spheres with blocks and holes

A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal $3$-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013.

math.CO↗

arXiv2013

On Spaces of Infinitesimal Motions and Henneberg Extensions

We investigate certain spaces of infinitesimal motions arising naturally in the rigidity theory of bar and joint frameworks. We prove some structure theorems for these spaces and as a consequence are able to deduce some special cases of a long standing conjecture of Graver, Tay and Whiteley concerning Henneberg extensions and generically rigid graphs.

math.MG↗

arXiv2009

Series Parallel Linkages

We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.

math.GT↗