New Techniques for Computing Geometric Index

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We introduce redgeneral new 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 red and allow the 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 B²× I in T. After that we prove that if a solid torus can be divided into solid chambers by meridional discs in a specific red(and often easy to obtain) way, then the geometric index can be easily computed.