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Register a new userIdeal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds
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Closed essential surfaces in a three-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebros module-theoretic interpretation of Culler-Shalen theory. As a corollary, we construct an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representations into PSL(2, C), resolving a question of Shanuel and Zhang.
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We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to its practicality, we run the algorithm over a comprehensive body of closed 3-manifolds and knot exteriors, yielding results that were not previously known. The algorithm derives from the original Jaco-Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. Our methods are relevant for other difficult computational problems in 3-manifold theory, such as the recognition problem for knots, links and 3-manifolds.
In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the $SL_2$-character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces) from ideal points of the $SL_n$-character variety for a natural number $n$ greater than or equal to 3. Further we verify that such a branched surface induces a nontrivial presentation of the 3-manifold group in terms of the fundamental group of a certain 2-dimensional complex of groups.
Extending Culler-Shalen theory, Hara and the second author presented a way to construct certain kinds of branched surfaces in a $3$-manifold from an ideal point of a curve in the $operatorname{SL}_n$-character variety. There exists an essential surface in some $3$-manifold known to be not detected in the classical $operatorname{SL}_2$-theory. We prove that every connected essential surface in a $3$-manifold is given by an ideal point of a rational curve in the $operatorname{SL}_n$-character variety for some $n$.
We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into $3$-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.
It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the K{u}hnel-Lutz conjecture is valid in dimension three for fields of odd characteristic. Next let $mathbb{F}$ be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is $mathbb{F}$-tight. For triangulated closed 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of an $mathbb{F}$-tight non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an $mathbb{F}$-tight triangulation of a closed 3-manifold has $n$ vertices and first Betti number $beta_1$, then $(n-4)(617n- 3861) leq 15444beta_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.
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