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I meet this problem when reading Rolfsen's Knots&Links. After giving 8 different definitions of linking number for knots in $S^3$, he left an exercise: Given disjoint PL knots $J$ and $K$ in $S^3=\partial D^4$, let $A$ and $B$ be 2-chains in $D^4$, such that $\partial A=J,\partial B=K$, assume $A$ and $B$ intersect transversally in a finite number of points, each point is given $1$ or $-1$ after making orientation conventions, then the weighted sum of intersection points is just the linking number for $J$ and $K$ in $S^3$.

This definition is heuristic and interesting, but I can't figure out how to understand this. Maybe an argument about integral or intersection in homology can give a proper explanation. Appreciation for any help!

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  • $\begingroup$ As a warm-up you could do the more immediate exercise: the linking number of two disjoint embedded $0$-spheres in $S^1$ can be computed by the intersection number of the corresponding cobounding intervals in $D^2$. $\endgroup$
    – Ryan Budney
    Commented Sep 19, 2023 at 15:27

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Here is a brief sketch. First, show that the intersection number between A and B is independent of the choice of specific chains. (In other words, if $A'$ and $B'$ are other 2-chains with the same properties, then $A \cdot B = A'\cdot B'$.)

Now you can compare this with one of the other definitions that take place in $S^3$ by choosing a particular $A, B$. For instance, if you like the version where you change crossings and keep track of signs, you can change crossings between $J$ and $K$ to make them isotopic to a split link $J' \cup K'$ (i.e. knots lying in disjoint balls). It's easy to construct a pair of cylinders in $S^3 \times I$ whose intersection number is exactly the number of crossing changes, counted with signs. Now think of $B^4 = S^3 \times I \cup D^4$ where $D^4$ is another disk. Then you can build chains $A$ and $B$ as the union of the aforementioned cylinders and disjoint disks (the cones on $J'$ and $K'$ in $D^4$, which can be taken to be disjoint.) Then $A\cdot B$ is the intersection number of the cylinders, which you've already proved to be the linking number.

Some care needs to be taken with conventions about orientations in order to get the signs right! There are also variations of this argument corresponding to the various definitions of the linking number as computed in $S^3$.

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  • $\begingroup$ So the sign of crossing point corresponds to the sign of intersection point of chains, when $J$ overpasses $K$, they become unkotted. Thanks for your answer, I think I get a better comprehension about this. $\endgroup$
    – Math Diego
    Commented Jun 23, 2022 at 12:08
  • $\begingroup$ You're welcome! Your understanding is correct, although I had in mind that after the crossing changes, J and K would become unlinked from each other, although they would still be individually knotted. $\endgroup$
    – Danny Ruberman
    Commented Jun 23, 2022 at 14:41
  • $\begingroup$ You are right, it can only be deduced that they are unlinked, whether they are unknotted is unknown. $\endgroup$
    – Math Diego
    Commented Jun 24, 2022 at 15:29

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