## The Computational Complexity of Knot and Link Problems (1997)

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### BibTeX

@MISC{Hass97thecomputational,

author = {Joel Hass and Jeffrey C. Lagarias},

title = {The Computational Complexity of Knot and Link Problems},

year = {1997}

}

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### Abstract

We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, splitting problem of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson. Keywords: knot theory, three-dimensional topolog...

### Citations

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Citation Context ...ious formulations of knot and link equivalence, and many other aspects of knot theory, we recommend the books [1], [6], [30]. An introduction to the notions of complexity which we use can be found in =-=[9]-=-,[41]. In order to study the computational complexity of knot and link problems, we must agree on a finite computational representation of a knot or link. There are two natural representations: a poly... |

1497 | Theory of Linear and Integer Programming - Schrijver - 1986 |

423 | mapping class groups - Birman, Braids - 1974 |

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Citation Context ...ich nodes can be written down in polynomial length and in which adjacency of nodes can be tested in polynomial space, the connectedness of the graph can be determined in polynomial space (see Savitch =-=[31]-=-). In step (5d), we compute the intersection number of ∂S with a marked meridian and longitude in ∂S. We trace the curve S, assigning an orientation to each segment, and keep a running total of the in... |

270 |
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Citation Context ...l time in the proof of Theorem 8.1. � Using Haken’s Theorem 5.4 instead of the Jaco-Tollefson result we still obtain the weaker result that the unknotting problem is in the polynomial hierarchy Σ p 2 =-=[38]-=-: One uses the same certificate as above, except for the proof that S is connected. For this step, we use a co-NP. oracle that answers the question: Is v in the Hilbert basis of CM? This problem is in... |

231 | Efficient planarity testing
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Citation Context ...s of several disjoint single crossing projections.) Let m denote the number of vertices of G, and call the m − n vertices added special vertices. Using the Hopcroft-Tarjan planarity testing algorithm =-=[17]-=- we can construct a planar embedding of G in time O(n log n). From this data we determine the planar faces of this embedding, and add extra edges to triangulate each face, thus obtaining a triangulate... |

203 |
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Citation Context ...ial AK(x) = 1, the same as the Alexander polynomial of the trivial knot. Another invariant that has been investigated with the same hope is the Jones polynomial JK(x) of a knot K, discovered by Jones =-=[22]-=- in 1985. In this case the complexity bound is less attractive: the Jones polynomial for links (a generalization of the Jones polynomial for knots) is #P-hard and in FP #P (see Jaeger, Vertigan and We... |

162 |
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Citation Context ...ass are said to be unknotted or trivial knots. We may remark at this point that while it is “intuitively clear” that there are non-trivial knots, it is not at all obvious how to prove this. Stillwell =-=[37]-=- traces the mathematical notion of knot back to a paper of A. T. Vandermonde in 1771; the first convincing proof of the non-triviality of a knot seems to be due to Max Dehn [8] in 1910. There are a gr... |

154 | How to draw a planar graph on a grid - Fraysseix, Pach, et al. - 1990 |

98 | New points of view in knot theory
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Citation Context ...Jones polynomial has led to the discovery of a great number of new knot and link invariants, including Vassiliev invariants and invariants associated to topological quantum field theories, see Birman =-=[4]-=- and Sawin [32]. The exact ability of these invariants to distinguish knot types has not been determined. A different approach to the problems of recognizing unknottedness and deciding knot equivalenc... |

78 | Topological invariants of knots and links - Alexander - 1928 |

70 |
The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
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Citation Context ...alence class of the knot unchanged. For more details on piecewise-linear topology, the various formulations of knot and link equivalence, and many other aspects of knot theory, we recommend the books =-=[1]-=-, [6], [30]. An introduction to the notions of complexity which we use can be found in [9],[41]. In order to study the computational complexity of knot and link problems, we must agree on a finite com... |

62 |
How to draw a planar graph on a grid, Combinatorica 10
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Citation Context ...iangulated planar graph G ′ , in time O(n). The graph G ′ has m vertices and 2m − 5 bounded triangular faces, and the unbounded face is also a triangle. It was shown by de Frajsseix, Pach and Pollack =-=[7]-=- that there exists a planar embedding of G ′ whose vertices v(G ′ ) lie in the plane z = −1, in the grid {(x, y, −1) : 0 ≤ x, y ≤ 10n − 1 ; x, y ∈ Z} , (19) in which all edges of G ′ are straight line... |

62 |
Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten
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(Show Context)
Citation Context ...edness and deciding knot equivalence eventually led to decision procedures. This approach is based on the study of normal surfaces in 3-manifolds (defined in Section 3), which was initiated by Kneser =-=[23]-=- in 1929. In the 1950’s Haken developed the theory of normal surfaces, and obtained a decision procedure for unknottedness in 1961. Haken considered compact orientable 3-manifolds. Schubert [34] exten... |

47 | Knot Theory and Its Applications - Murasugi - 1996 |

46 |
An algorithm to decide if a 3-manifold is a Haken manifold, Topology 23
- Jaco, Oertel
- 1984
(Show Context)
Citation Context ...es (incompressible surfaces), eventually resulting in a collection of 3-balls. Knot complements are examples of Haken manifolds. It gives a procedure to decide if two Haken manifolds are homeomorphic =-=[18]-=-. Recent work of Jaco-Oertel [18] and Jaco-Tollefson [20] further simplified some of these algorithms. Apart from these decidability results, there appear to be no explicit complexity bounds, either u... |

46 |
Algorithms for the complete decomposition of a closed 3-manifold
- Jaco, Tollefson
- 1995
(Show Context)
Citation Context ...ollection of 3-balls. Knot complements are examples of Haken manifolds. It gives a procedure to decide if two Haken manifolds are homeomorphic [18]. Recent work of Jaco-Oertel [18] and Jaco-Tollefson =-=[20]-=- further simplified some of these algorithms. Apart from these decidability results, there appear to be no explicit complexity bounds, either upper or lower, for any of the three problems that we stud... |

44 |
Über das Geschlecht von
- Seifert
- 1935
(Show Context)
Citation Context ...is in NP. Another generalization of the unknotting problem concerns the genus g(K) of a knot K. This is an integer assoicated to each knot, which is invariant under isotopy. It was defined by Seifert =-=[36]-=- in 1935; an informal account of the definition follows. Given a knot K, consider the class S(K) of all orientable spanning surfaces for K; that is, embedded orientable surfaces that have K as their b... |

39 |
A±ne structures in 3-manifolds V. The triangulation theorem and Hauptvermutung
- Moise
- 1952
(Show Context)
Citation Context ...a polygonal knot or link. This paper considers PL-knots and links. Given this restriction, we can without further loss of generality restrict our attention to the piecewise-linear settings (see Moise =-=[24]-=-). A regular projection of a knot or link is an orthogonal projection into a plane (say z = 0) that contains only finitely many multiple points, each of which is a 7sdouble point with transverse cross... |

39 |
Recursive unsolvability of group theoretic problems
- Rabin
- 1958
(Show Context)
Citation Context ... isomorphic to the infinite cyclic group. During the 1950’s it was shown that many such decision problems for finitely presented groups, not necessarily arising from knots, are undecidable (see Rabin =-=[28]-=-, for example), thus blocking 5sthis avenue of progress. The avenue has been traversed in the reverse direction, however: there are decision procedures for restricted classes of finitely presented gro... |

37 | C Lagarias, The number of Reidemeister moves needed for unknotting
- Hass, J
(Show Context)
Citation Context ...nce of Reidemeister moves that convert it to the trivial knot diagram. In this sense the unknotting problem is a purely combinatorial problem, though with no obvious bound on the number of steps, see =-=[12]-=-, [42]. 4 Unknottedness Criterion Our approach to solving the unknotting problem, based on that of Haken, relies on the following criterion for unknottedness: A PL-knot K embedded in R 3 is unknotted ... |

37 | Hilbert bases, Caratheodory’s Theorem and combinatorial optimization - Sebö - 1990 |

36 | On Dehn’s lemma and the asphericity of knots - Papakyriakopoulos - 1957 |

33 |
Residual finiteness for 3-manifolds, Combinatorial group theory and topology
- Hempel
- 1984
(Show Context)
Citation Context ...lexity bounds. One interesting question is whether the unknotting problem is in co-NP. Thurston’s geometrization theorem for Haken manifolds can be used to show that knot groups are residually finite =-=[16]-=-. It follows that a non-trivial knot has a representation into a finite permutation group with non-cyclic image. Unfortunately no way is yet known to bound the size of this group; if the number of sym... |

30 |
Theorie der normalflachen, ein isotopiekriterium für den kreisknoten
- Haken
- 1961
(Show Context)
Citation Context ... [41]—[43] for more information on this problem. The main result of this paper is the following. Theorem 1.1 The unknotting problem is in NP. The unknotting problem was shown to be decidable by Haken =-=[10]-=-; the result was announced in 1954, and the proof published in 1961. From then until now, we know of no strengthening of Haken’s decision procedure to give an explicit complexity bound. We present suc... |

24 |
The Classi of Knots and 3{Dimensional Spaces
- Hemion
- 1992
(Show Context)
Citation Context ...roblem [40]: Problem: KNOT EQUIVALENCE PROBLEM Instance: Two link diagrams D1 and D2. Question: Are D1 and D2 knot diagrams of equivalent knots? The final step in this program was completed by Hemion =-=[14]-=- in 1979. This program actually solves a more general decision problem, concerning a large class of 3-manifolds, now called Haken manifolds, which can be cut into “simpler” 6spieces along certain surf... |

21 | Knots in self-avoiding walks - Sumners, Whittington - 1988 |

19 |
Über die Topologie des dreidimensionalen Raumes
- Dehn
- 1910
(Show Context)
Citation Context ... prove this. Stillwell [37] traces the mathematical notion of knot back to a paper of A. T. Vandermonde in 1771; the first convincing proof of the non-triviality of a knot seems to be due to Max Dehn =-=[8]-=- in 1910. There are a great many alternative formulations of the notion of knot equivalence. Here are some. 1. One can consider sequences of elementary moves, which are very simple isotopies that move... |

18 | Bestimmung der Primfaktorzerlegung von Verkettungen - Schubert - 1961 |

17 |
quantum groups and TQFTs
- Sawin, Links
- 1996
(Show Context)
Citation Context ...al has led to the discovery of a great number of new knot and link invariants, including Vassiliev invariants and invariants associated to topological quantum field theories, see Birman [4] and Sawin =-=[32]-=-. The exact ability of these invariants to distinguish knot types has not been determined. A different approach to the problems of recognizing unknottedness and deciding knot equivalence eventually le... |

15 |
Recent results on sufficiently large 3–manifolds, from: “Algebraic and geometric topology
- Waldhausen
- 1978
(Show Context)
Citation Context ...cedure to decide the knot genus problem and related problems on arbitrary compact 3-manifolds with boundary. Haken also outlined an approach via normal surfaces to decide the knot equivalence problem =-=[40]-=-: Problem: KNOT EQUIVALENCE PROBLEM Instance: Two link diagrams D1 and D2. Question: Are D1 and D2 knot diagrams of equivalent knots? The final step in this program was completed by Hemion [14] in 197... |

14 |
The computational complexity of knot and link problems, preprint
- Hass, Lagarias, et al.
- 1998
(Show Context)
Citation Context ...2) and space O(n2 ), on a Turing machine. The results in this paper were announced in the Proceedings of the 38th Annual Symposium on Foundations of Computer Science in Miami Florida in October, 1997 =-=[13]-=-. 2 Historical Background Recognizing whether two knots are equivalent has been one of the motivating problems of knot theory. A great deal of effort has been devoted to a quest for algorithms for rec... |

13 | Über das Geschlecht von Knoten - Seifert - 1935 |

11 | On the Computational Complexity of the Jones and Tutte - Jaeger, Vertigan, et al. - 1990 |

11 | Complexity: Knots - Welsh - 1993 |

9 |
Knots in random walks
- Pippenger
- 1989
(Show Context)
Citation Context ...ce; a sequence of moves (up, down, north, south, east, west) that traverse the polygon, returning to the starting point without visiting any other point twice. (This formulation was used by Pippenger =-=[27]-=- and Sumners and Whittington [39] to show that “almost all” long self-avoiding polygons are non-trivially knotted.) The size of a polygonal representation L is the number of edges in L; its input leng... |

9 | A trivial knot whose spanning disks have exponential size - Snoeyink - 1990 |

8 |
Topological Invariants of Knots and
- Alexander
- 1928
(Show Context)
Citation Context ...ity by finding an invariant of the knot that (1) can be computed easily and (2) assumes some particular value only for the trivial knot. (Here invariant means invariant under isotopy.) Thus Alexander =-=[2]-=- defined in 1928 an invariant AK(x) (a polynomial in the indeterminate x) of the knot K that can be computed in polynomial time. Unfortunately, it turns out that many nontrivial knots have Alexander p... |

7 |
Recognizing the unknot, preprint
- Birman
- 1997
(Show Context)
Citation Context ...aches to knot and link algorithms include methods related to Thurston’s geometrization program for 3-manifolds (see [11] for a survey) and methods based on encoding knots as braids (see Birman-Hirsch =-=[5]-=-). These approaches currently have unknown complexity bounds. Our results are obtained using a version of normal surface theory as developed by Jaco-Rubinstein [19]. Among other things we show that Ha... |

7 |
Knots and braids: some algorithmic questions
- Welsh
- 1991
(Show Context)
Citation Context ...putational problem of recognizing unknotted polygons as follows: Problem: UNKNOTTING PROBLEM Instance: A link diagram D. Question: Is D a knot diagram that represents the trivial knot? See Welsh [41]—=-=[43]-=- for more information on this problem. The main result of this paper is the following. Theorem 1.1 The unknotting problem is in NP. The unknotting problem was shown to be decidable by Haken [10]; the ... |

7 | PL equivariant surgery and invariant decompositions of 3– manifolds - Jaco, Rubinstein - 1989 |

6 | Algorithms for recognizing knots and 3-manifolds - Hass - 1998 |

4 |
On Dehn’s Lemma and the Asphericity of
- Papakyriakopoulos
- 1957
(Show Context)
Citation Context ...hat a knot is trivial if and only if the corresponding group is infinite cyclic. The proof of what is still known as “Dehn’s Lemma” had a gap, which remained until filled by Papakyriakopoulos in 1957 =-=[26]-=-. A consequence is the criterion that a curve is knotted if and only if the fundamental group of its complement is nonabelian. Dehn also posed the question of deciding whether a finitely presented gro... |

4 |
The complexity of knots, in: Quo Vadis, Graph Theory
- Welsh
- 1993
(Show Context)
Citation Context ... Reidemeister moves that convert it to the trivial knot diagram. In this sense the unknotting problem is a purely combinatorial problem, though with no obvious bound on the number of steps, see [12], =-=[42]-=-. 4 Unknottedness Criterion Our approach to solving the unknotting problem, based on that of Haken, relies on the following criterion for unknottedness: A PL-knot K embedded in R 3 is unknotted if and... |

4 | Lectures on Three-manifold Topology, CBMS Regional - Jaco - 1977 |

3 |
Bestimmung der Primfactor zerlegung von Verkettungen
- Schubert
- 1961
(Show Context)
Citation Context ...ows. Problem: SPLITTING PROBLEM Instance: A link diagram D. Question: Is the link represented by D splittable? The splitting problem was shown to be decidable by Haken [10] in 1961, see also Schubert =-=[34]-=-. We establish the following result. Theorem 1.2 The splitting problem is in NP. Another generalization of the unknotting problem concerns the genus g(K) of a knot K. This is an integer assoicated to ... |

2 |
Knots in Self-avoiding
- Sumners, Whittington
- 1988
(Show Context)
Citation Context ..., north, south, east, west) that traverse the polygon, returning to the starting point without visiting any other point twice. (This formulation was used by Pippenger [27] and Sumners and Whittington =-=[39]-=- to show that “almost all” long self-avoiding polygons are non-trivially knotted.) The size of a polygonal representation L is the number of edges in L; its input length is the number of bits needed t... |

1 | English translation - Grenzgeb, Bd - 1932 |

1 | Lagarias, "The number of Reidemeister moves needed for unknotting", preprint - Hass, C |

1 | Residual finiteness for 3-manifolds", Combinatorial group theory and topology, 379--396 - Hempel - 1987 |