The and Fixed-Parameter Tractability

Adele Jackson

Supervised by Dr Benjamin Burton

Australian National University

Vacation Research Scholarships are funded jointly by the Department of Education and Training and the Australian Mathematical Sciences Institute. 1 Introduction

Informally, the recognition problem is to, given a , recognise whether or not it can be untangled. A substantial open question in computational is whether or not there exists a polynomial time to answer this question. We investigated the question of whether unknot recognition is fixed-parameter tractable (FPT) in the cutwidth of a of the manifold, which would be trivially implied by unknot recognition being in P. In this report, we give an exposition of unknot recognition being in NP, and describe the approaches we explored in trying to show unknot recognition is FPT.

2 The unknot recognition problem

1 3 Definition 2.1. A knot K is a smooth embedding fK : S → S . We can write this knot as an injective continuous map K : S1 → S3.

We say that two , K1 and K2, are equivalent if there exists some continuous 3 3 3 3 map f : S × [0, 1] → S such that f(s, 0) = s, f(s, t0): S → S for fixed t0 and 3 3 s ∈ So is a homeomorphism on S , and f ◦ K1 = K2. Then f is an ambient isotopy taking K1 to K2. Then the unknot recognition problem is, given a knot, to check whether it is equivalent to the circle. To answer this with a computation, we give another criterion for a knot being unknottable.

3 A criterion for unknot recognition

Theorem 3.1 ([3]). A knot K can be unknotted if and only if there exists a disk D with smooth boundary embedded in S3, such that its boundary, ∂D, is K traversed once. Alternately, we can view this disk as living in the complement of the knot, by setting up a criteria for this disk that does not directly reference traversing the knot’s boundary. 3 Definition 3.2. The knot complement manifold of a knot K, MK , is S − {a solid torus neighbourhood of K}. Definition 3.3. An essential disk in a manifold M is a disk D embedded in M such that the boundary of D lies in the boundary of M, and D cannot be homotoped into ∂M while holding ∂D fixed.

2 Figure 1: A solid torus with (a) a non-essential disk, and (b) an essential disk.

For example, disk A in Figure 1 is not essential, as it can be homotoped onto the section of the boundary of the manifold cut out by ∂A, while B is essential.

Theorem 3.4 ([3]). A knot K can be unknotted if and only if MK , the knot com- plement manifold, contains an essential disk.

So now we can express the unknot recognition problem as, given a knot comple- ment manifold, checking whether or not that manifold contains an essential disk. We wish to describe how one would do this, and assess how effective that algorithm is.

4 Computational complexity

When given an algorithm for a problem, we can describe how fast it is by how its running time grows as the size of the input n grows. This is effective as it does not vary with the hardware, language or details of implementation used to run the algorithm. Let the maximum number of steps taken for the algorithm to run for an input size n be M(n). We write that an algorithm is O(f(n)), for f a function of M(n) the input size, if lim supn→∞ f(n) < ∞. Complexity classes let us classify how difficult problems are. A decision problem is in class P if there is some algorithm to solve the problem that is O(P (n)) for P (n) a polynomial function. A decision problem is in NP if, if for some input the answer is ‘yes’, we can give a certificate to prove this is correct that can be verified in polynomial time. For example, consider the decision problem of checking whether there exists a timetable with no clashes for some set of rooms, students, lecturers and classes. This

3 Figure 2: A one tetrahedron triangulation of the solid torus. is a difficult problem to solve. However, if such a timetable does exist, the certificate that consists of that timetable can be checked in polynomial time to indeed have no clashes. Unknot recognition is known to be in NP, as we will show in Section 5.

5 Normal surfaces

To check whether or not a given knot complement manifold contains an essential disk, we need a way of concisely describing surfaces in a manifold. Normal surfaces are one method of doing this. Note that we can triangulate the knot complement manifold – that is, we can divide it into tetrahedra. For example, the knot complement manifold of S1 in S3 is again a solid torus. We can triangulate this using one tetrahedron by Figure 2, with face (023) joined to face (312).

Definition 5.1. A normal surface in a triangulated 3-manifold M is a surface S embedded in M, with boundary on the boundary of M, such that the intersection of S with any tetrahedron in the triangulation is a finite union of triangles and quadrilaterals.

For example, Figure 3 shows a normal surface within the triangulated torus, and how it looks in a drawing of the solid torus. The normal surface caps off the one vertex on the boundary of the torus. Given a normal surface S in a triangulation of M, we can describe this surface by listing how many of the four triangle types and three types of quadrilaterals it

4 Figure 3: A normal surface, (a) presented as triangles in the triangulated solid torus, and (b) drawn in the solid torus.

5 forms in each tetrahedron. For example, the surface in Figure 3 consists of one of each of the types of triangles in the one tetrahedron. For a surface S in a triangulation consisting of t tetrahedron, this defines a function v : normal surfaces → Z7t giving a vector describing each surface. For example, the surface in Figure 3 can be described by the vector (1, 1, 1, 1, 0, 0, 0), as it has one of each triangle and no quadrilaterals. This description is defined for a fixed ordering of tetrahedron and ordering of types of triangle and quadrilateral.

Theorem 5.2 ([3]). Any vector v ∈ Z7t describes a valid (possibly disconnected) normal surface S so long as:

1) all entries in v are non-negative,

2) each tetrahedron contains only one type of quadrilateral (as different types of quadrilateral must intersect), and

3) if two faces are connected in the triangulation, the edges intersecting these faces must match up.

The non-negativity and matching conditions define a cone of potential surfaces in Z7t, while the quadrilateral condition picks out a certain lattice of points within this cone. Then we can define an integer basis for this lattice.

Definition 5.3. We call a normal surface S fundamental if it cannot be written as the sum of two other normal surfaces. That is, if there exist no other surfaces S0, S00 such that v(S) = v(S0) + v(S00).

Now, it remains to show that normal surfaces are a useful way of detecting an essential disk in a knot complement manifold.

Theorem 5.4 ([5], strengthening [3]). If MK is a knot complement manifold, with an associated triangulation, which contains an essential disk, then it contains an essential disk that is a fundamental normal surface.

So if we can check all normal surfaces for an essential disk, we can solve the unknot recognition problem.

6 Unknot recognition is in NP

Hass, Lagarias and Pippenger used this approach to show that unknot recognition is in NP, by a bound on the size of the fundamental normal surfaces.

6 Theorem 6.1 ([4]). Let M be a triangulated compact 3-manifold, that contains t tetrahedra in the triangulation. Then the entries of the vector describing any funda- mental normal surface in M are bounded by t27t+2 − 1.

This then gives the following algorithm for checking whether a given knot K is unknottable, with input size n the number of crossings in K:

1) Produce a triangulation of the knot complement manifold MK . This can be done in polynomial time in n to have a number of tetrahedra t that is bounded by a polynomial function of n.

2) Generate all possible descriptions of fundamental normal surfaces in MK – that is, all vectors in Z7t with entries between 0 and t27t+2 − 1. 3) For each of these vectors, check if:

a) the vector satisfies the quadrilateral and matching conditions, so is a valid normal surface, b) the vector is a disk, for which it is sufficient to check the surface has one puncture and Euler characteristic 1 as all fundamental normal surfaces are connected, and c) the vector describes an essential disk, which is a condition on the ho- mology class of the boundary of the surface.

4) If one of these vectors did describe an essential disk, the knot is the unknot; otherwise, it is not.

All of these steps, aside from looking at every possible fundamental normal sur- face, take polynomial time in the crossing number of the original knot. Now, if the knot is equivalent to the unknot, we can give a polynomial time certificate for this: describe the triangulation and give the vector describing the essential disk. The size of the vector is bounded by the product of the number of entries and the logarithm 7t+2 2 of the maximum entry size, which is 7t log2(t2 −1) which is O(t ). Then checking the triangulation is correct and that the vector does indeed describe an essential disk are both polynomial time in t which is itself a polynomial function of n. This gives:

Theorem 6.2 ([4]). Unknot recognition is in NP.

7 This simple algorithm can be substantially improved on. The approach of con- sidering all vectors with entries up to t27t+2 − 1, of which there are O(t7t249t2+14t), is unneccessary. The number of candidates for the essential disk was reduced in the same paper to O(128t) [4], and has more recently been reduced to O(15t) [1]. We also have another result on the complexity of unknot recognition, that

Theorem 6.3 ([7]). Knot recognition is in NP. Equivalently, unknot recognition is in co-NP.

As unknot recognition is in both NP and co-NP, it is a good candidate to be in P. An intermediate step to showing unknot recognition to be in P is to show that it is fixed-parameter tractable.

Definition 6.4. A problem is fixed-parameter tractable (FPT) in a variable (that is not the input size) if, letting the input size be n and the variable be k, the problem can be solved in O(P (n)f(k)) for P (n) a polynomial function.

For example, the vertex cover problem – answering whether, for a given graph with n vertices, there exists a subset of vertices of size at most k such that every edge has at least one end in the subset – is FPT in k. (There is an algorithm that is O(2kn).) Trivially, if a decision problem is in P it is FPT in any variable k, as its complexity can be written as O(P (n)k0). So finding that unknot recognition is FPT would be further encouragement that it is potentially in P.

7 Approaches to showing unknot recognition is fixed-parameter tractable

We primarily explored showing that unknot recognition is fixed-parameter tractable in the cutwidth of the dual graph of the triangulation.

Definition 7.1. For some ordering o in the set of orderings O of the vertices V of a graph, and some fixed vertex v, let c(o, v) be the number of edges that pass from vertices after v in the ordering to v and the vertices before it. Then the cutwidth of a graph is mino∈O maxv∈V c(o, v). For example, the cutwidth of the dual graph of any layered solid torus is 2. (See [6] Section 4.2 for a discussion of layered solid tori, and for easily generating triangulations of them.)

8 The idea was to build surfaces as we built the triangulation, extending them each time a new tetrahedron was added, such that the size of the set of surfaces we were keeping track of at any point was FPT. Additionally, all that is required is a FPT description of the surfaces, rather than a FPT set of surfaces themselves, so long as this description gives enough information to be able to check if anything in the described set is the essential disk. We focused on finding a way of describing all normal surfaces that are punctured spheres. Note that a disk is a sphere with one puncture. A surface is a punctured sphere if it has no genus and is orientable. Equivalently, a surface is a punctured sphere if it has p disconnected boundary components, and Euler characteristic 2 − p. Note that we can always build punctured spheres out of pieces that all have no genus and orientable, so a description of all the punctured spheres in a triangulated manifold should be enough information to allow us to generate a description of all the punctured spheres in that manifold with one tetrahedron added without having to examine all of the original manifold.

Describing punctured spheres as sums of normal surfaces We looked at describing punctured spheres as sums of the fundamental normal sur- faces. For example, consider the 1,2,3-layered torus, depicted in Figure 2 and triangu- lated by one tetrahedron with face (023) joined to face (312). This manifold has four fundamental normal surfaces: α, a Moebius band that caps an edge with Euler characteristic χ = 0, β, an orientable surface that caps another edge with χ = 0, γ, the essential disk with χ = 1, and δ, the vertex with χ = 1. Now, there are two types of surfaces that have p punctures and χ = 2 − p. These are aα + bβ such that gcd(2a, b) = 2 and aα + γ with a even. However, this description has a substantial problem: the number of fundamental normal surfaces grows exponentially with t, the number of tetrahedra. We considered describing the fundamental surfaces by their Euler characteristic and the number of times they cross each edge on the boundary, which is sufficient to determine whether a sum of surfaces is a punctured sphere. However, the number of crossings can also grow exponentially with t, so while this reduces the number of surface descriptions, it does not make it FPT. This approach also has another issue: though we can guarantee that sums of fundamental normal surfaces satisfy the non-negative and matching conditions, they may have intersecting quadrilaterals (violating condition 2 in Theorem 5.2) so not be valid normal surfaces. This can happen in two ways, producing either immersed

9 or singular surfaces rather than embedded ones.

Figure 4: A singular surface, with the singular point at the intersection of the surface and the axis line.

Definition 7.2. We say a surface is embedded if there is an bijective continuous map from a region of R2 to the surface. A surface is immersed if this map is not injective, but if every point in the domain has a neighbourhood such that a restriction of the map to this neighbourhood is injective. A surface is singular if there is a point without such a neighbourhood.

For example, the surface in the 1,2,3-layered torus given by 2α + γ is a singular surface, seen in Figure 4. As singular and immersed surfaces are not normal surfaces, they are not candi- dates for being the surface describing the essential disk, but are rather more infor- mation than we need to track.

Describing a superset of the embedded punctured spheres Determining whether a normal surface vector describes an embedded surface is diffi- cult and expensive, so tracking only embedded punctured spheres is likely infeasible.

10 Instead, we attempted to describe a set of punctured spheres containing the set of em- bedded punctured spheres. Having some singular and immersed punctured spheres in this set is not an issue, by Dehn’s Lemma.

Theorem 7.3 (Conjectured by Dehn, 1910; proved Papakyriakopoulos, 1957). Let M be a 3-manifold, with f : S1 → M a continuous map taking ∂D to ∂M, and with f restricted to a neighbourhood of ∂D injective. Then there exists f 0 : S1 → M an injective continuous map, such that f 0 and f are identical on a neighbourhood of the boundary of the disk. (That is, suppose we have a disk in a 3-manifold with singularity on the interior of the disk. Then there exists a properly embedded disk in the 3-manifold with the same boundary.)

So if, looking at a description of punctured spheres in a manifold, the essential disk candidate we pick out is actually self-intersecting, we still have proof that there exists a properly embedded essential disk, as the singularity will occur in the intersection of quadrilaterals which occurs in the interior of tetrahedra. Unfortunately, even describing this smaller set of punctured spheres can be diffi- cult. Adding on a new tetrahedron that connects to the existing manifold at only one face and extending a set of punctured spheres through this tetrahedron is straightfor- ward. For each punctured sphere, we can take all non-intersecting ways of continuing its edges that hit the face that is connected to the new tetrahedron. The interest- ing cases, then, are adding on a new tetrahedron that connects on multiple faces, which is equivalent to adding a new tetrahedron on one face, then connecting pairs of unmatched faces of the manifold. However, when this operation creates a ‘bridge’ in the manifold, it can allow a large, difficult to describe family of punctured spheres to appear. For example, in the torus, all torus knots can be thickened to an annulus, and there is a distinct for every pair of coprime integers. Additionally, in some cases, joining even two faces of disconnected manifolds (so not producing this bridge) can produce an infinite number of punctured spheres from a finite number in each manifold. So this approach, too, does not look useful.

Future directions Given that these direct approaches to describing punctured spheres have been un- successful, a promising future direction for research would be to look at the algebraic structure given by punctured spheres. We can consider punctured spheres as sur- faces determined by the boundaries of their punctures p1, . . . , pn, with a punctured ±1 ±1 ±1 sphere satisfying p1 p2 ··· pn = 1. This structure is simpler than the fundamental

11 group, as the puncture boundaries are always disjoint. However, not all such lists of boundaries describe a punctured sphere (for example, consider a manifold made of two disconnected pieces, and a boundary in each of them). Using combinatorial group theory to study this structure would bring a new set of tools to the problem.

12 Acknowledgements

I would like to thank my supervisor, Dr Ben Burton, for his continual support and encouragement, and for taking the time from his summer to supervise a student from another university. I would also like to thank AMSI for funding the project, the organisers of the MATRIX introductory workshop on topology in December 2016 for allowing me to attend, and the participants in the MATRIX workshop and the MSI workshop on low-dimensional topology at ANU in November for the opportunity to learn about current research in the field.

13 References

[1] Burton, B.A. 2011. Maximal admissable faces and asymptotic bounds for the normal surface solution space. J. Combin. Theory Ser. A 118(4), pp. 1410-1435.

[2] Burton, B.A., Budney, R., and Pettersson, W., et al., 1999-2016. Regina: Soft- ware for low-dimensional topology, http://regina-normal.github.io/.

[3] Haken, W. 1961. Theorie der Normalflachen, ein Isotopiekriterium f¨urden Kreisknoten. Acta Math 105, pp. 245-375.

[4] Hass, J., Lagarias, J.C., and Pippenger, N. 1999. The computational complexity of knot and link problems. J. ACM 46(2), pp. 185-211.

[5] Jaco, W., and Tollefson, J. L. 1995. for the complete decomposition of a closed 3-manifold. Illinois J. Math. 39, pp. 358-406.

[6] Jaco, W., and Rubinstein, J.H. 2006. Layered-triangulations of 3-manifolds. Preprint at https://arxiv.org/abs/math/0603601.

[7] Lackenby, M. 2016. The efficient certification of knottedness and Thurston norm. Preprint at http://people.maths.ox.ac.uk/lackenby/knp30316.pdf.

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