Cappell-Shaneson knot pairs with the same Alexander polynomial

Abstract

It is well known that for $m\geq 2$ there are at most two non-equivalent $m$-knots with diffeomorphic exterior. Such pair of knots will be called $\textit{ non-reflexive knot pair}$. A classical problem in topology is to determine all dimensions where such knot pairs exist. In 1976 Cappell and Shaneson gave a method of constructing non-reflexive knot pairs. In the present paper we construct an infinite family of new examples of Cappell-Shaneson knot pairs, and give examples of Cappell-Shaneson knot pairs that have the same Alexander polynomial but are inequivalent.

Figures (3)

• Figure 1: Orders of ideal class groups corresponding to $x^4+ax^3+(-2a-2)x^2+(a-1)x+1$.
• Figure 2: Orders of ideal class groups corresponding to $x^5+ax^4+(-2a-1)x^3+(2a+1)x^2+(-a+1)x-1$.
• Figure 3: Orders of ideal class groups corresponding to $x^6+(2a+1)x^5+(a^2-a-2)x^4+(-2a^2-2a-2)x^3+(a^2-a-1)x^2+(2a+1)x+1$.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 3.1: Gu-Jiang
• Theorem 3.2
• proof
• Definition 3.3