Cyclic branched covers of alternating knots and $L$-spaces

TL;DR

This paper proves that for genus one alternating knots, the three-fold cyclic branched cover is also an $L$-space, extending known results about double branched covers.

Contribution

It establishes that the three-fold cyclic branched cover of genus one alternating knots is an $L$-space, generalizing previous results on double branched covers.

Findings

Three-fold cyclic branched cover is an $L$-space for genus one alternating knots.

Extends known results from double to triple branched covers.

Provides new insights into the topology of branched covers of knots.

Abstract

For any alternating knot, it is known that the double branched cover of the $3$-sphere branched over the knot is an $L$-space. We show that the three-fold cyclic branched cover is also an $L$-space for any genus one alternating knot.