Orateur: Jin Gyo
Taek (KAIST)
Titre: Quadrisecant approximation of minimal polygonal knots.
Résumé:
By straightening small subarcs into line segments, every tame knot $K$ can
be approximated by a polygonal knot $K^\prime$, within its knot type, whose
vertices are on $K$. If the vertices of $K^\prime$ are not well distributed
along $K$, $K^\prime$ may not have the knot type of $K$.
It is known that every nontrivial knot has a quadrisecant. Given a knot, we
mark each intersection point of each of its quadrisecants. Replacing each
subarc between two nearby marked points with a straight line segment joining
them, we obtain a polygonal closed curve which we will call the quadrisecant
approximation of the given knot. We show that for any heptagonal figure
eight knot in general position, there are only six quadrisecants, and the
resulting quadrisecant approximation has the same knot type. Furthermore,
the resulting quadrisecant approximation has no new quadrisecants other than
those of the heptagonal figure eight knot. We also show that for each pair
$(p,q)$ of relatively prime integers satisfying $1 < p < q < 2p$,
there exists a minimal polygonal torus knot of type $(p,q)$ having only
finitely many quadrisecants whose quadrisecant approximation is of the same
knot type.
This is a joint work with Seojung Park.