Stably irreducible non-orientable knotted surfaces

Abstract

We give an elementary obstruction to reducibility for knotted surfaces in the four-sphere. As a new application, we construct stably irreducible non-orientable surfaces.

Figures (2)

• Figure 1: A handle diagram for a 4-manifold $X'$ with $H_2(X') = \mathbb{Z}$ which admits an involution and has fundamental group $T(2,3,7)$ can be obtained by attaching 2-handles along cables of the green curves. In particular, from left to right, the $(7,2), (3,2)$, and $(2,1)$ cables work. Any choice of framings works.
• Figure 2: Top left, we see the arc $\alpha$ for which band surgery goes from $P(-2,3,7)$ to $P(-2,3,7,n)$, where $n$ is the number of half-twists in the band. Top right, we isotope $P(-2,3,7)\cup\alpha$ into a convenient position. Observe then that $P(-2,3,7)$ is obtained from the horizontal unknot by tangle replacements along the blue and green arcs marked in the bottom left. By the Montesinos trick, $\Sigma_2(P(-2,3,7))$ is obtained the double cover of the horizontal unknot by surgery along the blue and green link in the bottom right. Surgery on this link yields $S^2(0;1/2,-1/3,-1/7)$, and we see that $\alpha$ lifts to a regular fiber. For more details on passing between branch sets and surgery descriptions, we find Bloom helpful.

Theorems & Definitions (8)

• Conjecture : Kinoshita conjecture, see KS Remark 3.7
• Theorem
• Corollary
• proof : Proof of the Corollary
• Proposition
• proof
• proof : Proof of the Theorem
• Remark