Bulletin of the
Korean Mathematical Society BKMS

For a knot in the $3$--sphere, the Upsilon invariant is a piecewise linear function defined on the interval $[0,2]$. It is known that this invariant of an L--space knot is the Legendre--Fenchel transform (or, convex conjugate) of a certain gap function derived from the Alexander polynomial. To recover an information lost in the Upsilon invariant, Kim and Livingston introduced the secondary Upsilon invariant. In this note, we prove that the secondary Upsilon invariant of an L--space knot is a concave conjugate of a restricted gap function. Also, a similar argument gives an alternative proof of the above fact that the Upsilon invariant of an L--space knot is a convex conjugate of a gap function.

Keywords: Secondary Upsilon invariant, Upsilon invariant, knot Floer complex, Legendre--Fenchel transform, L--space knot

MSC numbers: Primary 57K10; Secondary 57K18

Supported by: The author has been supported by JSPS KAKENHI Grant Number 20K03587.