Rational concordance of double twist knots

TL;DR

This work classifies the rational sliceness and rational concordance of the double twist knots $K_{m,n}$, proving that such knots are rationally slice precisely when $mn=0$ or $|m+n|c1$, and that no new torsion arises in the rational concordance group for this family. The authors combine a smooth-category obstruction from lattices on infinitely many prime-power branched covers with Donaldson's diagonalization theorem, together with topological-category obstructions via von Neumann $\rho$-invariants with complexity, to obtain a complete picture. They also determine the algebraic rational concordance of twist knots and provide topological obstructions showing non-sliceness in those cases, linking Alexander polynomial data to higher-order $\rho$-invariants. Collectively, the results pinpoint the precise rational concordance class of $K_{m,n}$ and illuminate the interplay between smooth and topological sliceness in this family as well as the behavior under cabling and branched covers.

Abstract

Double twist knots $K_{m, n}$ are known to be rationally slice if $mn = 0$, $n = -m\pm 1$, or $n = -m$. In this paper, we prove the converse. It is done by showing that infinitely many prime power-fold cyclic branched covers of the other cases do not bound a rational ball. Our rational ball obstruction is based on Donaldson's diagonalization theorem.

Figures (4)

• Figure 1: The twist knot $K_n$.
• Figure 2: The double twist knot $K_{m,n}$.
• Figure 3: $K_{m,n}$ for $n > 0$ bounds a smooth nullhomologous disk in $n\overline{\mathbb{CP}}^2$.
• Figure 4: A negative definite filling $W_p(m,n)$ of $\Sigma_p(K_{m,n})$ for $m < 0$, $n > 0$, and $p > 2$.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 2.1
• proof : Proof of Theorem \ref{['thm-B']}
• Theorem 3.1
• Lemma 3.2