The intersection graph of an orientable generic surface

Hadar, Doron Ben

doi:10.2140/agt.2017.17.1675

Cited by 1 publication

(14 citation statements)

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“…Since in this case 1 3 N ≤ K ≤ 8N 3 , it is common to use N , instead of K, as a measure to the size of the formula, and the problem remains NP-complete with regards to N as the size parameter.…”

Section: )mentioning

confidence: 99%

“…We now have a symmetric 4-sat formula, F ∨ , that is satisfiable iff F is satisfiable, and we can deduce the solutions of F from those of F ∨ using Remark 2.2.8 (1). In order to modify it into a (symmetric) 3-sat formula, we use the following lemma: Lemma 2.2.9.…”

Section: Symmetric 3-sat Formulasmentioning

confidence: 99%

“…In particular, the d + 1 elements of each tuple must all be distinct and, since the order of the elements is irrelevant, we assume that the numbers in each tuple appear in ascending order. For instance, the triangle with vertices 1, 4 and 6 will be written as (1,4,6) and not (4,1,6).…”

Section: A Generic Surface As a Data Type -The Basicsmentioning

confidence: 99%

“…As per the above, we will go over every simplex τ of dimension d > 0 and every non-empty sub-complex σ of τ (for instance, if τ = (1, 3, 4) these will be 1, 3, 4, (1, 3), (1,4) and (3,4)). We also go over every subset µ of σ, this time allowing the empty set and σ itself, and add ν = (τ \ σ) ∪ µ to St(σ).…”

Section: A Generic Surface As a Data Type -The Basicsmentioning

confidence: 99%

“…The answer depends solely on the first homology group H 1 (M ; Z), and whether M has a boundary. We note that the result of chapter 9 have been submitted for publication as the article [1].…”

Section: Introductionmentioning

confidence: 99%

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The Lifting Problem is NP Complete

Hadar¹

2016

Preprint

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Let M be a 3-manifold. Every knotted (embedded) surface in M × R can be moved via an ambient isotopy in such a way that its projection into M is a generic surface. A surface is generic if every point on it is either a regular, double or triple value -the transversal intersection of 1, 2 or 3 embedded surface sheets, or a "branch value" that look like Whitney's umbrella. We elaborate on this in Definition 3.1.1. The double values form arcs, and along each arc two long strips of surface intersect. In a knotted surface, the additional R coordinate distinguishes between the two strips. One of them must be "higher" than the other. We elaborate on this in Definition 3.1.3.The lifting problem is the problem of determining if a generic surface in M can occur as the M -projection of a knotted surface in 4-space in M × R. The main purpose of this thesis is to study the computational aspects of the lifting problem. We will prove that the problem is NP-complete, and devise an efficient algorithm that determines if a generic surface is liftable.A surface can be lifted iff one can choose, along each of the double arcs of the surface, which of the two intersecting surface strips is "higher" without arriving at some sort of obstruction. There are two obstructions that might occur. First, what locally looks like two distinct surface strips may globally "join" into one strip. We call a double arc in which the two surface strips join "non-trivial".We elaborate on this in Definition 3.2.1. A generic surface that has a non-trivial double arc cannot be lifted (see Lemma 3.2.2). If the surface has no non-trivial arcs, one can attempt to lift the surface by choosing which of the two strips at each arc is the higher one. i List of Figures3.1 How to draw a generic surface and / or broken surface diagram .

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Section: )mentioning

confidence: 99%

Section: Symmetric 3-sat Formulasmentioning

confidence: 99%