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## gt.geometric topology – Wanted: a nontrivial weakly inadmissible Heegaard diagram

If you may have any 3-manifold $Y$ with optimistic $b_1$, and any (universal) Heegaard diagram $mathcal{H} = (Sigma, alpha, beta)$ for it, you may place the basepoint $z$ in order that $(Sigma, alpha, beta,z)$ shouldn’t be admissible.

In truth, deem any province $P_0$ in $mathcal{H}$ whose border is a nontrivial sum of $alpha$– and $beta$-curves (this might breathe referred to as sporadic, if the habitual definition of periodicity did not require that the multiplicity of the basepoint breathe 0), and let $m$ breathe the minimal multiplicity of $P_0$, attained at a area $D$. Consider $P = P_0 – mcdot Sigma$, and place the basepoint $z$ in $D$. Clearly, $P$ is a nonzero sporadic province in $(Sigma, alpha, beta,z)$ which has solely non-negative multiplicities.

Here’s what occurs for the “standard” admissible-looking diagram for $S^1times S^2$ you advert to. (I’m linking to an SVG portray, that I am unable to metamorphose on the significance — if anybody desires to edit, be happy to do it). The “standard” sporadic province is the dissimilarity of the 2 bigonal areas. Adding the all torus and inserting the basepoint within the 0-multiplicity bigon yields the linked portray.

Colour signifies multiplicity: white means 0, lighter grey 1, darker grey 2.

**EDIT**: as identified by the OP in a observation under, the province above has index 2, whereas we would love to have an index-0 sporadic province. I cerebrate I’ve an instance for this phenomenon (once more, a diagram for $S^2times S^1$):

Consider the province $P_0$ proven within the portray under (yellow means multiplicity -1, white multiplicity 0, and grey multiplicity 1), and generator $x$ akin to the 2 thick black factors, and the inexperienced basepoint $z$. The index of $P_0$ as a province in $pi_2(x,x)$ is -2 (the Euler quantity is the Euler attribute of the doubly-pointed torus, which is -2), and the multiplicity at $x$ is zero. If you add the gross floor, you get a sporadic province of index 0 with solely non-negative multiplicities.

Notice that there’s a lot of redundancy, each within the variety of turbines and the genus of the diagram, however I would not plane attempt to formulate a query the place that is addressed.

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