Slicing knots in definite 4-manifolds

TL;DR

This work introduces and analyzes CP^2-slicing numbers for knots, including smooth and topological variants, to understand how knots bound disks in definite 4-manifolds. It develops lower bounds via the double branched cover $\Sigma_2(K)$ and half-integer surgery type lattices, and constructs knots with arbitrarily large but finite smooth CP^2-slicing numbers, as well as cases where smooth and topological slicings diverge. The paper then provides a topological upper bound in terms of Seifert data, and applies the framework to alternating and pretzel knots to obtain explicit values and sharp bounds, including families where $u_{\mathbb{CP}^2}(K)=k$ while $u_{\mathbb{CP}^2}^{top}(K)=1$. It also investigates the relationship between topological and smooth slicings, giving concrete examples where the topological invariant is zero but the smooth invariant is nonzero, and establishing genus-based constraints that partially delineate the landscape of CP^2-slicing numbers. Overall, the results advance intrinsic smooth obstructions to sliceness in CP^2 and related definite 4-manifolds and illuminate the interaction between concordance, branched covers, and lattice obstructions in 4-manifold topology.

Abstract

We study the $\mathbb{CP}^2$-slicing number of knots, i.e. the smallest $m\geq 0$ such that a knot $K\subseteq S^3$ bounds a properly embedded, null-homologous disk in a punctured connected sum $(\#^m\mathbb{CP}^2)^{\times}$. We give a lower bound on the smooth $\mathbb{CP}^2$-slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth $\mathbb{CP}^2$-slicing number. We also give an upper bound on the topological $\mathbb{CP}^2$-slicing number in terms of the Seifert form and find knots for which the smooth and topological $\mathbb{CP}^2$-slicing numbers are both finite, nonzero, and distinct.

Figures (16)

• Figure 2.1: Left: We begin with an algebraically zero collection of strands in the diagram for a knot $K\subseteq S^3$. Middle: Perform $+1$-framed Dehn surgery on an unknot $U$ in $S^3$, encircling the given strands. Right: In the resulting copy of $S^3$, the strands gain a full negative twist. Passing from left to right in this figure consists of adding a generalized positive crossing.
• Figure 2.2: The knot $J_m$ from \ref{['ex:CNbound-2bridge']}. Each box indicates the number of positive half-twists.
• Figure 2.3: A linear plumbing diagram for the $4$-manifold $W(p,q)$ with boundary $L(p,q)$. Here, $\frac{p}{q}$ has the continued fraction expansion shown in \ref{['eq:negative-cont-frac']}, with $a_1,a_2,\dots,a_n\geq 2$. Adjacent generators have intersection $+1$.
• Figure 3.1: Proof of \ref{['thm:beast']}. The knots $R$ and $K'$ are shown in black, and the link $\gamma_1\sqcup \gamma_2 \sqcup \cdots \sqcup \gamma_m$ in purple. We perform $+1$-framed Dehn surgery on $S^3$ along each $\gamma_i$, or equivalently, add generalized positive crossings to $R$ guided by $\gamma_1\sqcup \gamma_2 \sqcup \cdots \sqcup \gamma_m$. Compare with \ref{['fig:gencrossing']}.
• Figure 3.2: Proof of \ref{['thm:beast']} continued. Left: The ribbon disk $D_R$ for $R$ is glued on to $R\times [0,1]$ to produce a disk within $W\cup B^4$. Here $W$ is obtained from $S^3\times [0,1]$ by attaching $+1$-framed $2$-handles along $\gamma_i\times \{1\}\subseteq S^3\times \{1\}$, for each $i=1, \dots, m$. The case $m=1$ is pictured. Depicted in green is a $+1$-framed $2$-sphere obtained as the union of the core of the $2$-handle attached to $\gamma_1\times \{1\}$, the cylinder $\gamma_1\times [0,1]$, and a slice disk for $\gamma_1$ in $B^4$. Right: The slice disk $D_{K'}$ for $K'$ in $(\#^m \mathbb{CP}^2)^{\times}$ is shown. It is glued on to the concordance $C\subseteq S^3\times [0,1]$ from $K'$ to $K$ to produce a slice disk for $K$ in $(\#^m \mathbb{CP}^2)^{\times}$, The case $m=1$ is pictured, and the core $\mathbb{CP}^1\subseteq (\mathbb{CP}^2)^\times$ is shown in green. As indicated there is a diffeomorphism taking the picture on the left to the picture on the right.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Proposition 1.5
• Example 2.1
• Definition 2.2: CochranTweedy*Definition 2.7
• Lemma 2.3: CochranTweedy*Theorem 5.7
• Remark 2.4
• Proposition 2.5