Moscow--Petersburg seminar on low-dimensional mathematics
SPb, PDMI, 24.05.2002, 16:00--17:45, room 311

E.A. Kudryavtseva, D.L.Gon\c{c}alves and H.Zieschang

"Intersection index of curves on surfaces and
a new algorithm to find the minimal number of intersection points"

Let $\gamma_1,\gamma_2$ be two curves on a surface $S$ with fixed
endpoints on $\partial S$. We define the {\it homotopy intersection
index} $\lambda(\gamma_1,\gamma_2)$ of these curves taking values in
the group ring ${\bold Z}[\pi_1(S)]$ of the fundamental group of the
surface. As for the algebraic intersection index
$\langle \gamma_1,\gamma_2 \rangle \in \bold Z$, its value depends on
the homotopy classes $g_i=[\gamma_i]$, $i=1,2$ of curves only.
We show (theorem of Turaev) that the number of non-vanishing
coefficients of the element $\lambda(\gamma_1,\gamma_2)$ equals the
minimal number of intersection points of curves
$\tilde\gamma_1,\tilde\gamma_2$ over all curves of the homotopy
classes $g_1,g_2$.

We give a combinatorial algorithm calculating the minimal number of
intersection points for given elements $g_1,g_2 \in \pi_1(S)$. In
contrast to known algorithms (Reinhart, Zieschang, Chillingworth,
Birman-Series, Lustig,...) we find also the homotopy classes of the
intersection points of the curves $(\hat\gamma_1,\hat\gamma_2)$ of
homotopy classes $g_1,g_2$ and realizing the minimum.