A note on surfaces in $\mathbb{CP}^2$ and $\mathbb{CP}^2\# \mathbb{CP}^2$

TL;DR

The paper investigates the CP^2-genus of knots, $g_{ ext{CP}^2}(K)$, showing it is unbounded and contrasting it with its topological counterpart. It develops a relative genus framework in $( ext{CP}^2)^ imes$ and extends minimal genus questions to $ ext{CP}^2 ilde{ imes} ext{CP}^2$, using invariants from Heegaard Floer theory and Freedman–Quinn disk embedding to build knots with large genus and to analyze torus knots. A key result is that the naïve additive genus bound for classes in $ ext{CP}^2 ilde{ imes} ext{CP}^2$ can be arbitrarily far from the true genus, highlighting a substantial gap between smooth and topological slicings. The work also provides explicit constructions showing that the CP^2-genus can be large while the topological CP^2-genus remains small, with consequences for the minimal genus problem in $ ext{CP}^2 ilde{ imes} ext{CP}^2$ and related homology classes.

Abstract

In this brief note, we investigate the $\mathbb{CP}^2$-genus of knots, i.e. the least genus of a smooth, compact, orientable surface in $\mathbb{CP}^2\setminus \mathring{B^4}$ bounded by a knot in $S^3$. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the $\mathbb{CP}^2$-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in $\mathbb{CP}^2\# \mathbb{CP}^2$.

Figures (3)

• Figure 2.1: The figure on the left shows the torus knot $T_{a+b, a+b-1}$. A box with an integer denotes that many full twists (right-handed for positive, left-handed for negative integers). The box containing $-\tfrac{1}{a+b}$ indicates a negative fractional twist, i.e. the rightmost strand passes across to the extreme left, above all the other strands. The second figure shows how to change $a \cdot b$ crossings with a blow-up, at the expense of adding a full twist on each of the two bundles of parallel strands. The third figure shows a simplified diagram of the knot in the second figure, after one forgets the $(+1)$-framed $2$-handle. From the last figure it is clear that the knot in question is $T_{a, 2a-1} \# T_{b, 2b-1}$.
• Figure 2.2: The figure on the left shows the torus knot $T_{a+b+1, a+b}$. The second figure shows how to change $a \cdot b$ crossing with a blow-up, at the expense of adding a full twist on each of the two bundles of parallel strands. After simplifying the diagram (third figure), the knot is identified with $T_{a, 2a+1} \# T_{b, 2b+1} \# T_{1,0} = T_{a, 2a+1} \# T_{b, 2b+1}$. (Note the sign change in the fractional twist in the third figure.)
• Figure 3.1: Attaching a $(+1)$-framed 2-handle to $B^4$ along an unknot gives the standard handle decomposition for $(\mathbb{CP}^2)^\times$. In the resulting surgery description of $S^3$, the knot $K$ depicted above appears unknotted, though blowing down the $(+1)$-surgery curve to get the empty surgery diagram for $S^3$ reveals that $K=-T_{n,n-1}$. The image of the standard slice disk for the unknot in $B^4$ in $(\mathbb{CP}^2)^\times$ is the desired slice disk for $K$, and represents $n$-times the generator of relative second homology. The case $n=3$ is shown.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Lemma 2.1
• Proposition 2.2
• proof
• proof : Proof of \ref{['thm:top-v-smooth']}
• Remark 2.3
• Remark 2.4