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We prove that the homeomorphism problem for 2--manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.
We use technology from sutured manifold theory and the theory of Heegaard splittings to relate genus reducing crossing changes on knots in S^3 to twists on surfaces arising in circular Heegaard splittings for knot complements. In a separate paper, currently in preparation, we prove that these circular Heegaard splittings may be searched for algorithmically, and together our results imply that an algorithm to detect when two hyperbolic or fibered knots of different genus are related by a cross...
We introduce a generalized Anshel-Anshel-Goldfeld (AAG) key establishment protocol (KEP) for magmas. This leads to the foundation of non-associative public-key cryptography (PKC), generalizing the concept of non-commutative PKC. We show that left selfdistributive systems appear in a natural special case of a generalized AAG-KEP for magmas, and we propose, among others instances, concrete realizations using $f$-conjugacy in groups and shifted conjugacy in braid groups. We discuss the advantage...
In graph theory, Courcelle's theorem essentially states that, if an algorithmic problem can be formulated in monadic second-order logic, then it can be solved in linear time for graphs of bounded treewidth. We prove such a metatheorem for a general class of triangulations of arbitrary fixed dimension d, including all triangulated d-manifolds: if an algorithmic problem can be expressed in monadic second-order logic, then it can be solved in linear time for triangulations whose dual graphs have...
We present a new, practical algorithm to test whether a knot complement contains a closed essential surface. This property has important theoretical and algorithmic consequences; however, systematically testing it has until now been infeasibly slow, and current techniques only apply to specific families of knots. As a testament to its practicality, we run the algorithm over a comprehensive body of 2979 knots, including the two 20-crossing dodecahedral knots, yielding results that were not pre...
Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our proof uses a reduction from Boolean constraint satisfaction problems where every variable appears in a...
There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both the geometry and combinatorics of the triangulation. We prove that detecting taut angle structures is NP-complete, but also fixed-parameter tractable in the treewidth of the face pairing graph of the triangulation. These results have deeper implications: t...
Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds, through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand we prove that the erasability problem is W[P]-complete on...
The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm is known. We present three methods for computing crosscap number that offer varying trade-offs between precision and speed: (i) an algorithm based on Hilbert basis enumeration and (ii) an algorithm based on exact integer programming, both of which either com...
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