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Two transverse essential simple curves or proper arcs on a surface are in minimal position exactly when they form no bigon. For proper arcs, the proof uses the compactification of the universal cover to control their ends at punctures.

A Dehn twist about a simple closed curve $\gamma$ in an oriented surface is a homeomorphism supported in an annulus around $\gamma$. In the coordinates and orientation convention of a right-handed Dehn twist, it has the formwhere $f$ is zero near one boundary circle and one near the other. It extends by the identity outside the annulus; subtracting $2\pi f(r)$ gives its inverse. Changing the interpolation without changing the end values gives the same mapping class.

For essential simple closed curves $\alpha ,\gamma$ and every integer $n$,Thus $i(\alpha ,\gamma )>0$ forces all iterates to have distinct isotopy classes. In minimal position of curves or arcs, each crossing strand winds $n$ times through the other strands; the bigon criterion proves the counted intersections cannot be removed.^[ref]

On a torus, take curves representing a basis $a,b$ of first homology, with $a\cdot b=1$. Under the stated shear convention, $T_a^n(b)=b+na$, and its geometric intersection number with $b$ is $|n|$. A twist about a curve bounding an unpunctured disc is instead isotopic to the identity by the Alexander trick. Boundary-parallel twists can remain nontrivial when isotopies must fix the boundary pointwise.

A collection of essential simple closed curves fills a surface if every essential simple closed curve has positive geometric intersection number with at least one member. For such curves in minimal position of curves or arcs on a closed surface of positive genus, this is equivalent to every complementary component being a disc.

The Alexander method uses a stronger cutting condition when boundary or marked points are present: an Alexander system must cut the surface into discs with at most one marked point each. For example, a pair of pants has no essential nonperipheral simple closed curves, so the empty collection passes the curve-intersection test, but it does not cut the surface into discs. The boundary-relative version must therefore keep the stronger condition explicit.

The geometric intersection number $i(\alpha ,\beta )$ is the minimum number of transverse intersection points among representatives of the isotopy classes of $\alpha$ and $\beta$. Representatives in minimal position realize it.

A sufficient condition is that both collections are Alexander systems: within each collection the essential simple closed curves are pairwise nonisotopic, are in pairwise minimal position of curves or arcs, have no triple intersection points, and no three curves intersect pairwise. If ${\alpha }_i$ is isotopic to ${\beta }_i$ for every $i$, the simultaneous-isotopy lemma for Alexander systems gives an ambient isotopy $H_t$ with $H_1({\alpha }_i)={\beta }_i$ for all $i$. Successive applications of the bigon criterion prove the lemma while preserving the curves already matched.

The arcs $\alpha$ and $\beta$ bound a bigon if there are subarcs with common endpoints whose union is the boundary of an embedded disc and whose interiors are disjoint. They are in minimal position if $|\alpha \cap \beta |$ is the least possible intersection count among proper arcs isotopic to $\alpha$ and $\beta$ relative to their ends.