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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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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. –  Ryan Budney Jul 11 '12 at 16:44
    
Never thought of it this way! Will chew upon this. Thanks a ton! –  Aneesh Karthik C Jul 11 '12 at 16:58
    
@Budney, You can infact turn this into an answer. I'll accept it. –  Aneesh Karthik C May 31 '13 at 6:26
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up vote 3 down vote accepted

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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