Decreasing the binding number of an open book while increasing the genus of the pages

Question

Let $(B,\pi)$ be an open book decomposition of a closed, connected, oriented 3-manifold $M$ with odd (even) binding number and with pages of Euler characteristic $\chi$. Is it possible to define another open book decomposition $(B',\pi')$ of $M$ with binding number one (two) and with pages of the same Euler characteristic $\chi$? Basically, I am trying to replace each pair of boundary components with a genus in pages of $(B,\pi)$ to obtain $(B',\pi')$.

I believe that this is too good to be true, but I have no candidate for a counter-example. However, I think one thing that might work is to pick the contact structure $\xi$ on $M$ compatible with $(B,\pi)$, and looking for $(B',\pi')$ in the collection of all open book decompositions compatible with $\xi$. Maybe if $\xi$ satisfies some certain properties (e.g. tightness), we can detect such a $(B',\pi')$.

Is there any paper that (partially) answers this question or a similar one? Also, is there an obvious counter-example or proof for it (in the general case and in the case where $\xi$ is tight)?

Answer 1

What you hope for is not in general possible.

For example, the overtwisted contact structure $\xi_{-\frac{1}{2}}$ on $S^3$ with $d_3 = -\frac{1}{2}$ (ie. the overtwisted contact structure in the same homotopy class as $\xi_{\rm{std}}$) has a supporting open book which is a thrice-punctured sphere, by [Etnyre–Ozbagci, Lemma 5.5]. If we could replace this open book with one that has a once-punctured torus page, the binding would be a genus-1 fibred knot in $S^3$. Thus, the binding is either the right-handed trefoil, the left-handed trefoil, or the figure-eight knot, which support $\xi_{\rm{std}}$, $\xi_{\frac{3}{2}}$, and $\xi_{\frac{1}{2}}$, respectively.

Additionally, there is a tight contact structure $\xi$ on $L(3,1)$ supported by $(S_{0,3},\delta_1\delta_2\delta_3)$, where $\delta_i$ are positive Dehn twists about the three boundary components. If there were an open book decomposition $(S_{1,1}, \phi)$ supporting $\xi$, then the binding is a genus-1 fibred knot in $L(3,1)$. It's shown in [Baker] that $L(3,1)$ admits two equivalence classes of genus-1 fibred knots. Since the bindings of $(S_{1,1}, a^{-3}b^{-1})$ and $(S_{1,1}, a^{-3}b)$ are non-equivalent genus-1 fibred knots in $L(3,1)$, both supporting overtwisted contact structures, it follows that $\xi$ is not supported by a genus-1 open book with one binding component.

Furthermore, we can get similar counter-examples even while ignoring the contact structure. The open book $(S_{0,3},\delta_1\delta_2\delta_3^5)$ supports $L(11,9)$. By [Baker], this lens space does not have a genus-1 fibred knot, and so is not supported by an open book with genus-1 pages and connected binding.