# nLab torus knot

## Idea

A torus knot is one that can be drawn on the surface of a torus. Examples include the trefoil

and cinquefoil

knots.

## Definition

A knot $K$ is said to be a torus knot if it can be embedded in the surface of a torus, that is, we have the map $K:{S}^{1}\to {ℝ}^{3}$ factors through the embedding of some torus ${T}_{1}\cong {S}^{1}×{S}^{1}$ into ${ℝ}^{2}$;

${S}^{1}\stackrel{K}{\to }{T}_{1}\stackrel{\mathrm{embed}}{\to }{ℝ}^{3}$S^1\stackrel{K}{\to}T_1\stackrel{embed}{\to} \mathbb{R}^3

category: knot theory

Created on December 5, 2011 20:19:45 by Tim Porter (95.147.237.179)