Morse diagrams, Murasugi sums, and the mapping class group

Abstract: A combinatorial Morse structure encodes a mapping class for a surface with boundary, and the data may be efficiently represented via a Morse diagram. This diagram determines an open book decomposition of a 3-manifold, and hence, a contact structure on that 3-manifold. We examine how combinatorial Morse structures behave under the connect sum of open books, with particular attention paid to the case of negative stabilisation. This leads to a diagrammatic criterion for detecting overtwisted contact structures. Finally, in the case of open books with one-holed torus pages, we classify all the Morse diagrams associated to a fixed open book decomposition.

• The paper introduces a combinatorial framework using Morse diagrams to represent open book decompositions and detect overtwisted contact structures.
• It details a method for constructing Morse diagrams via Murasugi sums, linking diagram moves with Dehn twist factorizations in the mapping class group.
• The work classifies Morse diagrams for one-holed torus pages and offers algorithmic criteria that streamline investigations in 3-manifold contact topology.

Morse Diagrams, Murasugi Sums, and Their Relationship with the Mapping Class Group

Introduction

The paper presents a combinatorial framework for studying open book decompositions of $3$-manifolds and their associated contact structures, leveraging Morse diagrams as efficient, graphical representations of Morse structures. By analyzing the behavior of these diagrams under Murasugi sums and stabilizations, the authors establish results that connect the combinatorics of Morse diagrams with contact topology and the mapping class group of surface pages. Key implications include a diagrammatic criterion for the detection of overtwistedness in contact manifolds and a classification of all Morse diagrams corresponding to a given open book with a one-holed torus page.

Combinatorial Morse Structures and Morse Diagrams

The framework begins with the formalization of combinatorial Morse structures on a surface with boundary. These structures consist of sequences of handle structures connected by singular handle structures corresponding to arc slides. The monodromies arising from these sequences are encoded via Morse diagrams—planar or annular diagrams recording traces of co-core endpoints and their combinatorial evolution under the sequence of arc slides.

An essential point established is that the equivalence class of Morse diagrams, possibly under weaker relations, provides representations corresponding to open book decompositions, contact manifolds, or $3$-manifolds, depending on the level of granularity in equivalence relations considered.

Murasugi Sums and Their Diagrammatic Realization

The authors extend classical operations in $3$-manifold theory—namely, the Murasugi sum and connected sum of open books—to the domain of Morse diagrams via a canonical splicing operation. This construction produces a new Morse diagram for the connect sum of two open books directly from the diagrams of the original factors. The combinatorial description involves partitioning and rearranging the constituent rectangles of the original diagrams and systematically extending trace curves according to the starlike boundary patterns induced by Murasugi polygons.

Theorem: The splice of Morse diagrams $X^1$ and $X^2$ yields a Morse diagram for the connected sum open book $$(\Sigma_1 * \Sigma_2, \overline{\phi}^1\circ\overline{\phi}^2)$$.

This constructive approach provides a concrete, combinatorial realization of the connect sum operation at the level of Morse diagrams, in contrast to prior non-constructive existence results.

Stabilization, Overtwistedness, and Diagrammatic Criteria

Positive and negative stabilizations, corresponding to Murasugi sums with the right- or left-handed Hopf band, respectively, have distinct implications in contact topology. While positive stabilization preserves tightness, negative stabilization always produces overtwisted contact structures.

A key result is a diagrammatic characterization of overtwisted contact manifolds:

Theorem: A contact manifold is overtwisted if and only if it admits a Morse diagram featuring a left-veering handle.

A left-veering handle is formally defined as a pair of trace curves, one vertical and one monotonic non-increasing in horizontal coordinate (up to teleportation events). This provides an entirely diagrammatic, combinatorial method for certifying overtwistedness, extending earlier criteria such as those based on non-right-veering monodromy [HKM].

Comparison with Goodman's sobering arcs and the construction of overtwisted disks highlights the strength of the Morse diagram approach: the left-veering handle yields a transverse overtwisted disk, not merely a violation of the Thurston-Bennequin inequality at the Legendrian boundary.

Morse Diagrams and the Mapping Class Group: One-Holed Torus Case

For open books with one-holed torus pages, the authors fully analyze the equivalence of Morse diagrams under a finite set of combinatorial "Morse moves." These moves encapsulate all possible relations between diagrams compatible with a fixed open book decomposition.

Theorem: Two Morse diagrams encode the same open book if and only if they can be related by a sequence of the specified Morse moves.

Furthermore, this structure enables an explicit translation between sequences of handle slides in the Morse diagram and Dehn twist factorizations of the monodromy in the mapping class group. The construction utilizes the standard presentation of $\mathrm{Mod}(\Sigma_{1,1})$, allowing for concise correspondence between diagram moves and mapping class relations.

This classification not only clarifies the combinatorial possibilities for Morse diagrams on the one-holed torus but suggests an alternative, more tractable combinatorial approach to handling mapping class factorizations for larger classes of surfaces—potentially mitigating the complexities of standard mapping class presentations.

Implications and Future Directions

Practically, the combinatorial approach to encoding open books and contact structures via Morse diagrams offers algorithmic and computational advantages for investigating $3$-manifold contact topology. Diagrammatic criteria for overtwistedness and the explicit classification in the torus case provide automated detection and enumeration methodologies. Theoretically, the correspondence between Morse diagrams and mapping class factorization opens avenues for re-examining the structure of open books as combinatorially tractable entities, with direct implications for contact topological properties.

Future developments may extend the finite move classification to broader families of surfaces and examine the universality of the left-veering handle criterion. The combinatorial Morse structure formalism positions itself as a foundation for advances in both the efficient calculation of invariants (e.g., Heegaard Floer theoretic invariants) and the explicit construction of contact manifolds with prescribed properties.

Conclusion

This work presents a rigorous, combinatorial approach to the study of open book decompositions and contact $3$-manifolds via Morse diagrams. The authors establish concrete, algorithmic procedures for constructing Morse diagrams under connect sum and stabilization, provide a precise diagrammatic overtwistedness criterion, and classify Morse diagrams for one-holed torus pages. These results both deepen the connection between diagrammatic, combinatorial, and mapping class group perspectives and suggest further combinatorial approaches to longstanding problems in low-dimensional topology and contact geometry.