New Techniques for Computing Geometric Index

We introduce new general techniques for computing the geometric index of a link L in the interior of a solid torus T. These techniques simplify and unify previous ad hoc methods used to compute the geometric index in specific examples and allow a simple computation of geometric index for new examples where the index was not previously known. The geometric index measures the minimum number of times any meridional disc of T must intersect L. It is related to the algebraic index in the sense that adding up signed intersections of an interior simple closed curve C in T with a meridional disc gives +/- the algebraic index of C in T. One key idea is introducing the notion of geometric index for solid chambers of the form B-2 x I in T. We prove that if a solid torus can be divided into solid chambers by meridional discs in a specific (and often easy to obtain) way, then the geometric index can be easily computed.