Tight fibered knots and band sums

TL;DR

This paper proves that non-trivial band sums cannot produce tight fibered knots unless they are connected sums, implying such sums do not yield L-space knots or knots with finite surgeries, with examples illustrating the distinctions.

Contribution

It provides a short proof linking tight fibered knots and band sums, showing non-trivial band sums do not produce tight fibered or L-space knots, with illustrative examples.

Findings

Non-trivial band sums of tight fibered knots are connected sums.

Such band sums do not produce L-space knots.

Examples show non-trivial band sums can produce prime knots that are fibered or quasipositive but not both.

Abstract

We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yields an L-space knot. Consequently, any knot obtained by a non-trivial band sum cannot admit a finite surgery. For context, we exhibit two examples of non-trivial band sums of tight fibered knots producing prime knots: one is fibered but not tight, and the other is strongly quasipositive but not fibered.