I'm currently reading Knots, Links, Braids, and 3-Manifolds by V. V. Prasolov and A. B. Sossinsky. I have trouble understanding the following picture. enter image description here The dashed line denotes a trefoil whose tubular neighborhood is to be cut out, and the thickened line denotes a longitude with frame number 1 around the dashed trefoil. Therefore, after pasting back the solid torus, the meridian would be pasted onto the thickened line therefore bounding a disk in the new 3-manifold. Hence we would have a nontrivial relation in the fundamental group. What I don't understand is the specific order of elements depicted in the circle on the right.

Any help is greatly appreciated. Thank you very much.


1 Answer 1


It turns out that the order is actually not that important: choose a vertex on the circle to start from and a direction to travel in, and then read letters. If the orientation of the edge disagrees with your direction of travel, invert the letter (so save yourself some trouble by traveling counter-clockwise).

If you and I happen to make identical choices except for a choice of starting vertex, then the words we write down will differ by conjugation (in the free group on the set of generators). If additionally we disagreed about the direction of travel, then one of us will have to invert our word. However, the relations we each come up with will be true in both presentations, because they only differ by conjugation and inversion in the free group.


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