Hyperbolic L-space knots not concordant to algebraic knots

TL;DR

The paper targets the question of where hyperbolic L-space knots sit in the knot concordance group relative to algebraic knots. It constructs an infinite family $K_n$ of hyperbolic L-space knots via braids and Montesinos-trick surgery, computes their Alexander polynomials and $\Upsilon$ invariants, and uses the non-integrality of $-3\int_0^2Υ_{K_n}(t)\,dt$ as an obstruction to expressing them as linear combinations of algebraic knots. It also proves the $K_n$ are not algebraic by their semigroups and shows a subsequence is linearly independent in the topological concordance group, with additional examples listed from Baker–Kegel. The results identify concrete non-algebraic obstructions among L-space knots and deepen understanding of concordance structure for L-space knots. These findings have implications for distinguishing L-space knot classes and for the study of plane-curve singularity invariants in Floer-theoretic knot theory.

Abstract

We construct hyperbolic L-space knots that are not concordant to any linear combination of algebraic knots.

Figures (10)

• Figure 1: Left: The surgery description of $K_n$. For the link $L=K\cup C_1\cup C_2$, perform $(-1)$-surgery on $C_1$ and $(-\frac{1}{n+1})$-surgery on $C_2$. Then $K$ is changed to $K_n$. Right: The knot $K_1=m211$.
• Figure 2: The Upsilon function $\Upsilon_{K_n}(t)$. Left shows the case $n=1$, and Right shows the case $n\ge 2$.
• Figure 3: The knot $K_1$. Left: the staircase. Right: the graph of the gap function. The red lines are piecewise-linear convex supporting functions, used to compute the Fenchel--Legendre transform.
• Figure 4: After $(-1)$-surgery, the surgery diagram gives $(4n+24)$-surgery on $K_n$. Each rectangle box contains right-handed $(n+1)$-half twists.
• Figure 5: The double branched cover of $S^3$ branched over this link gives the resulting manifold of $(4n+24)$-surgery on $K_n$.

Theorems & Definitions (32)

• Theorem 1.1
• proof : Plan of the proof
• Definition 2.1: see Wang
• Definition 3.1
• Theorem 3.4
• proof
• Lemma 3.5
• Corollary 3.7: see Tange
• Proposition 3.8: see OZ
• Proposition 3.9