Plumbed 3-Manifolds and Neumann Moves

TL;DR

This paper presents an explicit algorithm to convert weakly negative definite plumbing trees into negative definite ones using Neumann moves, combining plumbing calculus with eigenvalue extraction techniques.

Contribution

It introduces a constructive method that systematically eliminates positive eigenvalues from plumbing trees, enhancing understanding of 3-manifold plumbing structures.

Findings

Every weakly negative definite plumbing tree can be transformed into a negative definite one.

Positive eigenvalues are supported on linear branches and can be eliminated systematically.

The algorithm combines plumbing calculus with eigenvalue diagonalization techniques.

Abstract

We give a constructive proof that every weakly negative definite plumbing tree can be transformed into a negative definite one by a finite sequence of Neumann moves. The argument combines Neumann's plumbing calculus with the diagonalization algorithm of Duchon, Eisenbud, and Neumann, which extracts the eigenvalues of the framing matrix directly from the combinatorics of the tree. We show that any positive eigenvalues are supported on linear branches and can be eliminated systematically via controlled applications of Neumann moves. This provides an explicit algorithm reducing weakly negative definite plumbing trees to negative definite ones.