New Techniques for Computing Geometric Index

Abstract

We introduce \textcolor{red}{general} 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 \textcolor{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 $\pm$ the algebraic index of $C$ in $T$. One key idea is introducing the notion of geometric index for solid chambers of the form $B^2\times I$ in $T$. After that we prove that if a solid torus can be divided into solid chambers by meridional discs in a specific \textcolor{red}{(and often easy to obtain)} way, then the geometric index can be easily computed.