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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!

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    $\begingroup$ 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. $\endgroup$
    – Ryan Budney
    Jul 11, 2012 at 16:44
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    $\begingroup$ Never thought of it this way! Will chew upon this. Thanks a ton! $\endgroup$
    – Karthik C
    Jul 11, 2012 at 16:58

2 Answers 2

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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 separating torus.

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  • $\begingroup$ If this is the case, then even if each torus deformation retracts onto each of these cylinders it doesn’t deformation retract onto X_mn, the original required space, since the two « cylinders » don’t intersect in the required circle no? $\endgroup$
    – Little Narwhal
    Feb 13 at 19:54
  • $\begingroup$ @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. $\endgroup$
    – porridgemathematics
    Mar 11 at 14:42
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To expound a bit:

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.

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