Hyperbolic L-space knots and their Upsilon invariants

Abstract

For a knot in the 3-sphere, the Upsilon invariant is a piecewise linear function defined on the interval [0,2]. For an L-space knot, the Upsilon invariant is determined only by the Alexander polynomial of the knot. We exhibit infinitely many pairs of hyperbolic L-space knots such that two knots of each pair have distinct Alexander polynomials, so they are not concordant, but share the same Upsilon invariant. Conversely, we examine the restorability of the Alexander polynomial of an L-space knot from the Upsilon invariant through the Legendre-Fenchel transformation.

Figures (17)

• Figure 1: The knots $K_1$ and $K_2$ are the images of $K$ after performing $(-1/n)$--surgery on $C_1$ and $(-1/2)$--surgery on $C_2$.
• Figure 2: The graph of the gap function $2J(-m)$ for the $(-2,3,7)$--pretzel knot and its convex hull (broken line).
• Figure 3: The graph of the gap function of $K_1$ with $n=1$.
• Figure 4: The parts of the graph of a gap function. The broken lines show the parts of convex hull with slope $s$.
• Figure 5: The deformation for $K_1$. Each rectangle box contains horizontal right-handed half-twists with indicated number.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.3
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.3
• proof