# The isotopy problem for the phase tropical line

## TL;DR

This paper proves that the complex line defined by 1+z1+z2=0 and its phase tropical counterpart are topologically equivalent through an explicit isotopy construction.

## Contribution

It provides the first explicit isotopy between a complex algebraic line and its phase tropical version.

## Key findings

- Established isotopy between complex line and phase tropical line.
- Constructed explicit isotopy map.
- Confirmed topological equivalence of the two lines.

## Abstract

Let $H$ be the complex line $1+z_{1}+z_{2}=0$ in $(\mathbb{C}^{*})^{2}$ and $H_{trop}$ the associated phase tropical line. We show that $H$ and $H_{trop}$ are isotopic as topological submanifolds, by explicitly constructing an isotopy map.