I ask here the question since it hasn't been answered in Math Stack Exchange.

I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one exercise. I have spoken with a professor and he encouraged me to ask here or look for it in books since the exercise (which i will insert below) is related to 3-manifold theory, which isn't covered in the book and I have never studied it before. Here it goes what is explained on the book:





I have managed to solve Exercise 21.29 and to prove Proposition 21.31 in an alternative way using Mayer-Vietoris (with some help: here). My problem is in Exercise 21.30 since I have no idea of how to start or how to deal with it.

In the beginning I thought I could use Exercise 21.29 and see both [C] and [J] as the images of [C] by the isomorphism I showed in Exercise 21.29. And then all I had to do is to show that there are the same number of "windings" that "under-crossings", 2 of each. However, I think this procedure is not rigorous.

I am afraid I can't show no more progress on the problem besides the idea above. I have been thinking for a few days but no more useful ideas have come to my mind.

Any help would be appreciated, even if it consists on pointing me to a book or reference where I can find something related. And please I would really appreciate detailed explanations if you have any idea since this is the first time I deal with something like this.

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    $\begingroup$ Hint: Draw a Seifert surface for the knot (an embedded oriented surface in $S^3$ with boundary equal to $K$) and count the signed intersections. The Seifert surface represents the dual of the generator of $H_1(X)$ under the intersection pairing. $\endgroup$ – Adam May 4 '16 at 12:18
  • $\begingroup$ First of all, thanks for your time @Adam. What do you mean by signed intersections? Intersections with what? The boundary of $X$? Thanks in advance. $\endgroup$ – D1811994 May 5 '16 at 19:29
  • $\begingroup$ If you fix orientations on the surface and on the curves, each transverse intersection counts as a +1 if the orientation on $S^3$ given by the orientation "orientation of curve, orientation of surface" agrees with the standard one. Otherwise, it counts as a -1. Heuristically, it is the number of times the curve goes up through the surface minus the number of times it goes down through the surface. $\endgroup$ – Adam May 5 '16 at 21:22

It suffices to find a surface $F$ in the knot complement so that the boundary of $F$ is homologous to $[J] + [C]$. That will prove that $[J] + [C]$ is zero in $H_1(X)$. Which surface should you check first?

It may be helpful show that $[J] + [C]$ is homologous to a curve on $\partial X$. That curve will essentially follow $J$, but "winds one more time" (three times instead of twice).

  • $\begingroup$ First of all thanks for your answer and your time Sam Nead. Second, the first surface that comes to my mind is the punctured torus, which is in fact homeomorphic to the Seifert surface for this knot. Is my intuition working fine? And I am afraid I don't fully understand your second paragraph. Why It may be helpful show that $[J]+[C]$ is homologous to a curve on $\partialX$? I'm sorry for this questions, I suppose that this should be obvious but I have never studied 3-manifolds nor Knot Theory so I have never had exposition to this techniques. $\endgroup$ – D1811994 May 4 '16 at 21:14
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    $\begingroup$ How does the Seifert surface (with its boundary on the knot) meet $\partial X$? $\endgroup$ – Sam Nead May 5 '16 at 17:36
  • $\begingroup$ It meets $\partial X$ at the crossings of the knot? I'm afraid I have no idea about this. Do you know any reference where this is explained in detail? $\endgroup$ – D1811994 May 5 '16 at 19:38
  • $\begingroup$ I learned this material out of Rolfsen's book "Knots and links" and the book "Knots, links, braids, and three-manifolds" by Prasolov and Sossinsky. $\endgroup$ – Sam Nead May 7 '16 at 8:45

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