Michael Polyak
Whitney's formulas for curves on surfaces.
Tue, 11/03/2009 - 16:01 — SomNambulAuthors: Michael Polyak, Yurii Burman
Subjects: Geometric Topology
AbstractThe classical Whitney formula relates the number of times an oriented plane
curve cuts itself to its rotation number and the index of a base point. In this
paper we generalize Whitney's formula to curves on an oriented punctured
surface. To define analogs of the rotation number and the index of a base point
of a curve, we fix an arbitrary vector field on the surface. Similar formulas
are obtained for non-based curves.Minimal sets of Reidemeister moves.
Mon, 08/24/2009 - 13:10 — SomNambulAuthors: Michael Polyak
Subjects: Geometric Topology
AbstractIt is well known that any two diagrams representing the same oriented link
are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending
on orientations of fragments involved in the moves, one may distinguish 4
different versions of each of the O1 and O2 moves, and 8 versions of the O3
move. We introduce a minimal generating set of oriented Reidemeister moves,
which includes two moves of types O1 and O2, and only one move of type O3. We
then consider other sets of moves and show that only few of them generate all
Reidemeister moves.
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