New family of hyperbolic knots whose Upsilon invariants are convex

Abstract

The Upsilon invariant of a knot is a concordance invariant derived from knot Floer homology theory. It is a piecewise linear continuous function defined on the interval $[0,2]$. Borodzik and Hedden gave a question asking for which knots the Upsilon invariant is a convex function. It is known that the Upsilon invariant of any $L$-space knot, and a Floer thin knot after taking its mirror image, if necessary, as well, is convex. Also, we can make infinitely many knots whose Upsilon invariants are convex by the connected sum operation. In this paper, we construct infinitely many mutually non-concordant hyperbolic knots whose Upsilon invariants are convex. To calculate the full knot Floer complex, we make use of a combinatorial method for $(1,1)$-knots.

Figures (25)

• Figure 1: The case where $q=3k+1$. $\sigma_1$ and $\sigma_2$ are standard generators of the $3$-braid group. The box with number $n$ indicates $n$ right handed vertical full-twists.
• Figure 2: The case where $q=3k+2$. The box with number $-n$ indicates $n$ left handed vertical full-twists.
• Figure 3: The full knot Floer complex ${\rm CFK}^{\infty}(T(3,4))$ of $T(3,4)$. As an $\mathbb{F}_2[U,U^{-1}]$--module, there are five generators (vertices). Actually, by the action of $U$, complexes of the same shape are lined up in the upper right and lower left. However, we draw only the complex which has generators with Maslov grading $0$ (white vertices). Also, the differentials are represented by line segments, for example, $\partial[b,1,3]=[a,0,3]+[c,1,1]$.
• Figure 4: A box complex. The homological cycle is $d$ or $b+c$. But both are boundary cycles, so this complex is acyclic.
• Figure 5: The full knot Floer complex ${\rm CFK}^{\infty}(K_n^{(3,3k+1)})$. This complex consists of the staircase complex, which is consistent with the complex of ${\rm CFK}^{\infty}(T(3,3k+1))$, and box complexes. The vertices of box complexes are actually on the grid, but are drawn slightly displaced.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Proposition 2.2: HoLi
• Lemma 3.1
• proof
• proof : Proof of (1) in Theorem \ref{['mainthm1']}
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4