Homogeneous braids are visually prime

TL;DR

The paper proves that closures of homogeneous braids are visually prime by translating the problem into an open-book primeness criterion: if a summing region is essential and the pieces satisfy veering conditions, the resulting open book has no fixed essential arcs. It then represents the Seifert surface of a non-split homogeneous braid as an iterated Murasugi sum of strictly veering pieces arranged in a tree, and applies the criterion inductively to deduce primeness. The authors also show that primeness is not preserved by all plumbing operations: trefoil plumbings can create fixed arcs and yield non-prime bindings, whereas figure-eight plumbings preserve primeness; they further prove arborescent fibered links are prime. Overall, the work provides a robust framework linking open-book theory, veeringness, and Murasugi sums to visually detect primeness in a broad class of fibered links, and it clarifies how specific plumbing operations influence primeness. The results advance Cromwell’s program by delivering a concrete, diagram-independent criterion for primeness in the homogeneous-braid setting and by exposing the limits of primeness preservation under common plumbing moves.

Abstract

We show that closures of homogeneous braids are visually prime, addressing a question of Cromwell. The key technical tool for the proof is the following criterion concerning primeness of open books, which we consider to be of independent interest. For open books of 3-manifolds the property of having no fixed essential arcs is preserved under essential Murasugi sums with a strictly right-veering open book, if the plumbing region of the original open book veers to the left. We also provide examples of open books in S^3 demonstrating that primeness is not necessarily preserved under essential Murasugi sum, in fact not even under stabilizations a.k.a. Hopf plumbings. Furthermore, we find that trefoil plumbings need not preserve primeness. In contrast, we establish that figure-eight knot plumbings do preserve primeness.

Figures (14)

• Figure 1: Top left: the knot $6_3$ and the stabilizing arc $a$ (red). Top right: the result of the stabilization (band being plumbed in blue). Bottom: the result, after an isotopy, is seen to be a non-prime link: the connected sum of the prime fibered knot $8_{20}$ and a Hopf link.
• Figure 2: Left: $D(\beta_{\mathrm{comp}})$ (top) with a decomposition circle (dotted, red) and $D(\beta_{\mathrm{prime}})$ (bottom). Right: Corresponding Seifert surfaces.
• Figure 3: The fiber surfaces $\Sigma_1=F(2,-3)$, $\Sigma_2=F(2,3)$, $\Sigma_3=F(2,5)$, $\Sigma_4=F(2,-3)$ (from top to bottom), together with summing regions $P_1$, $P_2$, and $P_3$ such that Murasugi summing $\Sigma_2$ to $\Sigma_1$ using $P_1$, summing $\Sigma_3$ to the result via $P_2$, and summing $\Sigma_4$ to that result via $P_3$ yields $\Sigma(\beta_{\mathrm{comp}})$ (left) and $\Sigma(\beta_{\mathrm{prime}})$ (right).
• Figure 4: The fiber surfaces $\Sigma_1=F(2,-3)$ (top), $\Sigma_2$ (middle), $\Sigma_3=F(2,-3)$ (bottom), where $\Sigma_2$ arose by Murasugi summing $F(2,5)$ to $F(2,3)$ using the summing region $P_2$ from \ref{['fig:murasugisforbetaprandbetadec']}.
• Figure 5: Model of $\Sigma$ in a ball-neighborhood of $P$ in the $3$--manifold associated to a $3$--Murasugi sum. The $6$--gon $P$ along which we are summing is shaded in gray, and lies on a plane $E$. The page $\Sigma_1$ lies below $E$, and above it, lies the page $\Sigma_2$. In the notation of \ref{['rmk:visualMS']}, in case $M_1\cong M_2\cong S^3$, up to diffeomorphism $M$ can be identified with the one point compactification of $\mathbb{R}^3$ such that $S^2$ is the one-point compactification of $E$, $M_1\setminus \mathrm{int}(B_1)$ and $M_2\setminus \mathrm{int}(B_2)$ are the lower and upper half-spaces with boundary $E$, respectively, and the pages $\Sigma_i\subset M$ are surfaces in the said half-spaces.

Theorems & Definitions (36)

• Theorem 1.1: Homogeneous braids are visually prime
• Example 1.2
• Proposition 1.3
• Proposition 1.4
• Definition 3.1
• Remark 3.2
• Remark 3.3
• Lemma 3.4
• Example 3.5
• proof : Proof of \ref{['lemma:lemma1']}