Stable Knotted Strings

TL;DR

This paper analyzes the dynamics of relativistic closed strings with knotted topologies in Minkowski space, providing conditions for their periodicity, collapse, and the generation of complex knots from simple links.

Contribution

It offers the first comprehensive solution to the Cauchy problem for knotted relativistic strings, including stability and knot generation mechanisms.

Findings

Knotted strings can be time-periodic surfaces in spacetime.

A knotted string collapses to a link at quarter of its initial length.

Generation of complex knots from simple initial configurations is possible.

Abstract

We solve the Cauchy problem for the relativistic closed string in Minkowski space $M^{3+1}$, including the cases where the initial data has a knot like topology. We give the general conditions for the world sheet of a closed knotted string to be a time periodic surface. In the particular case of zero initial string velocity the period of the world sheet is proportional to half the length ($\ell$) of the initial string and a knotted string always collapses to a link for $t=\ell/4$. Relativistic closed strings are dynamically evolving or pulsating structures in spacetime, and knotted or unknotted like structures remain stable over time. The generation of arbitrary $n$-fold knots, starting with an initial simple link configuration with non zero velocity is possible.