# Regarding fundamental group computation of the complement of a torus knot in $S^3$

This is regarding the fundamental group computation of the complement of a toral knot in $S^3$ in Hatcher's algebraic topology book. See page 48. I have understood till the stage where the cross section of the torus minus the knot deformation-retracts to the radial segments as the arrows indicate. What is not clear is "Letting $x$ vary, these radial segments then trace out a copy of the mapping cylinder $X_m$ in the first solid torus."

I tried imagining this with a simple cases like the trefoil knot, but can't fathom this statement. Any help would be greatly appreciated!

• It helps to think of the 3 sphere as the union of two solid tori. The circles hatcher uses in the mapping cylinder construction are the cores of the solid torus plus a curve parallel to the knot in the separatingtorus. Jul 11, 2012 at 16:44
• Never thought of it this way! Will chew upon this. Thanks a ton! Jul 11, 2012 at 16:58

• @LittleNarwhal the two cylinders should intersect on the curve parallel to the knot in the separating torus - which is a copy of $S^1$. This is also the 'domain' of either mapping cylinder, so it is precisely what deformation retracts onto the respective ends. Mar 11 at 14:42
The center of the "Y" figure is a point of $$S^1$$ of the mapping cylinder. This center also wraps around the torus once as $$x$$ varies. However, the tips of the "Y" wrap around the torus $$m$$ times.