A complete list of lens spaces constructed by Dehn surgery I

TL;DR

This paper proves the completeness of Berge's list of doubly primitive knots producing lens spaces via Dehn surgery for cases where >1, and confirms the completeness of a table related to Poincare9 homology sphere surgeries.

Contribution

It establishes the completeness of Berge's classification of knots for >1 and verifies the completeness of a key table for Poincare9 homology sphere surgeries.

Findings

Berge's list of doubly primitive knots is complete for >1.

Table 6 in [8] is complete for Poincare9 homology sphere surgeries when >1.

Introduces the invariant  related to lens space surgery.

Abstract

Berge in [1] defined doubly primitive knots, which yield lens spaces by Dehn surgery. At the same paper he listed the knots into several types. In this paper we will prove the list is complete when $\tau>1$. The invariant $\tau$ is a quantity with regard to lens space surgery, which is defined in this paper. Furthermore at the same time we will also prove that Table~6 in [8] is complete as Poincar\'e homology sphere surgery when $\tau>1$.