论文标题
在模块化琼斯多项式上
On the modular Jones polynomial
论文作者
论文摘要
结理论的一个主要问题是决定琼斯多项式是否检测到结节。在本文中，我们研究了一个较弱的相关问题，即琼斯多项式是否减少了整数$ n $可以检测到UNNENNOT。众所周知，答案是$ n = 2^k $的负面，带有$ k \ geq 1 $和$ n = 3 $。在这里，我们证明，如果答案对约$ n $是负面的，那么对于任何$ k \ geq 1 $，对于$ n^k $来说是负的。特别是，对于任何$ k \ geq 1 $，我们构建了琼斯多项式的非平凡结〜$ 3^k $。
A major problem in knot theory is to decide whether the Jones polynomial detects the unknot. In this paper we study a weaker related problem, namely whether the Jones polynomial reduced modulo an integer $n$ detects the unknot. The answer is known to be negative for $n=2^k$ with $k\geq 1$ and $n=3$. Here we show that if the answer is negative for some $n$, then it is negative for $n^k$ with any $k\geq 1$. In particular, for any $k\geq 1$, we construct nontrivial knots whose Jones polynomial is trivial modulo~$3^k$.
