四割线逼近猜想的反例
Counterexamples to the quadrisecant approximation conjecture
摘要：
一个纽结的四割线是一条与纽结相交于四点的直线。如果一个纽结有200多个四割线，则可以用两个相邻割线点之间的线段来代替它们之间的每个子弧，以获得原始纽结的四割线近似。据证实，四割线近似总是与原始结具有相同结类型的结。我们证明了每种纽结类型都包含两个纽结，其中一个纽结的四割线近似有自相交，而另一个纽结的四割线近似是不同纽结类型的纽结。
英文摘要：
A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self intersections while the quadrisecant approximation of the other knot is a knot with different knot type.
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