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Let $K_1,K_2$ be two knots and $X_1, X_2$ be their knot complements. Let $K_3$ be sum of two knots and $X_3$ the resp. complement. I am tempted to thinking that there is some relationship between $X_i$'s (for example, trying to show $X_3$ can be obtained from subspaces of $X_1, X_2$.)

I see that connected sum of $n$ trefoil knots or even with Hopf links are fibered (from John Harer's "How to construct all fibered knots and links" lemma $2.1$). So a conjecture can be raised: ''Sum of fibered knots is fibered.''.

Is this true? And is this related to the relation between knot complement of knot sums above?

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First, welcome to MSE!

And yes, the connect sum of two fibered knots is fibered.

This can be seen relatively easily by considering the topological definition of connect sum. Place your two separate knots inside an $S^2$ each. Then pull a small arc with no knotting outside of the sphere. Then we create the connect sum by taking the interior of the two spheres and gluing them together along their boundary so that the two points where the knots intersect the sphere line up.

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Fibered knots are knots which have a Seifert Surface that can rotate around the knot. Since the knots originally had this property, then there was a small part of that Seifert Surface that would have been on the outside of those spheres which also did this. This was a disk with part of its boundary on the sphere's boundary. We just need to line these boundaries up on each sphere, which is always possible since two curves with common endpoints on a sphere are isotopic.

Hope this helps!

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    $\begingroup$ Something that helped me believe your argument was to imagine a summand as a long knot intersecting the green sphere in the north and south poles, where outside the green sphere the knot is just the z axis. We can arrange things so that a Seifert surface from the fibration is, outside the green sphere, a half-plane with boundary the z-axis, meeting the green sphere in geodesic from the north to sound pole. Somehow thinking about it this way made it clearer to me that the fibrations could be glued together -- I'll just leave this comment here in the unlikely case it's helpful to someone else! $\endgroup$ Commented May 20, 2021 at 17:31
  • $\begingroup$ Thanks for the hint! I initially try to glue the two knot complement directly but removing a part of tube from boundaries of each $X_1, X_2$ and glue the resp. Seifert surfaces along small disks near the segments (the one removed in knot sum), turns out I cannot show that the resultant space is $X_3$. Also thanks for Kyler Miller's interpretation of the construction! $\endgroup$
    – user928824
    Commented May 21, 2021 at 1:01

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