论文标题
戈迪安距离和完整的亚历山大邻居
Gordian Distance and Complete Alexander Neighbors
论文作者
论文摘要
如果每个可能的亚历山大多项式都通过一个结，我们将一个完整的亚历山大邻居称为“结” $ k $。尚不清楚是否存在一个具有非平凡亚历山大多项式的完整的亚历山大邻居。我们从拥有该特性的情况下消除了具有非平凡亚历山大多项式的无限结家族，并讨论未解决案例的可能策略。 此外，我们对结的决定因素使用条件，一个越过的交叉变化从一个结的第一结中进行了变化，以改善11和12个交叉结上的打结数字数据。 Lickorish引入了一个障碍物，这证明了相同的结果。但是，我们表明舔lickorish的障碍物并不能纳入决定因素的条件所带来的障碍物。
We call a knot $K$ a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from $K$. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases. Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo's unknotting number data on 11 and 12 crossing knots. Lickorish introduced an obstruction to unknotting number one which proves the same result. However, we show that Lickorish's obstruction does not subsume the obstruction coming from the condition on determinants.
