I have two questions that seem to be related.

I wonder if there is a user-friendly algorithm (starting from ribbon/slice presentation of knots/disks) for the construction of double (in general $p$-fold) coverings of $B^4$ along ribbon/slice disks.

Using the handle decomposition of ribbon/slice disks can we control the handle decomposition of double (in general $p$-fold) coverings of $B^4$ along these disks?


2 Answers 2


I would like to popularize the lecture notes of Brendan Owens about the double branched cover of knots in $S^3$ and of ribbon/slice disks in $B^4$.

He both discussed constructions and obstructions. Also, it includes the classical references listed by Professor Ruberman.

Owens, Brendan. "Knots and 4-manifolds." Winter Braids Lecture Notes 6 (2019): 1-26.

It is avaliable online here.


The first is in Akbulut-Kirby, Branched covers of surfaces in 4-manifolds. (Math. Ann. 252, 111-131 (1980). See the end of Section 3; it's basically explained via a single example (the square knot ribbon). The idea is to trade the ribbon intersections for extra handles in the complement. You can work out the answer to your second question from this.

The first is also given (with a bit of explanation) as exercise 6.3.5 (d)* in Gompf-Stipsicz. The * means that there is a solution in the back. It also appears as Exercise 11.8 (again, with a hint) in Akbulut's 4-manifold book.

  • $\begingroup$ Correction: 6.3.5. (d) $\endgroup$ Feb 9, 2022 at 16:31
  • 1
    $\begingroup$ @MaxSchumann Thanks; I corrected it in my answer. $\endgroup$ Feb 9, 2022 at 17:01

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