Hard Problems in 3-Manifold Topology
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1 Hard Problems in 3-Manifold Topology Einstein Workshop on Discrete Geometry and Topology Arnaud de Mesmay 1 Yo av Rieck 2 Eric Sedgwick 3 Martin Tancer 4 1 CNRS, GIPSA-Lab 2 University of Arkansas 3 DePaul University 4 Charles University Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
2 Embeddings in R d Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
3 Embed k d Problem: Embed k d Given a k-dimensional simplicial complex, does it admit a piecewise linear embedding in R d? Embed 1 2 is Graph Planarity Embed 2 3 : does this 2-complex embed in R 3? Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
4 Embed k d d always embeds k never embeds 7 Polynomially decidable - Hopcroft, Tarjan 1971 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
5 Embed k d d always embeds k never embeds?? Polynomially decidable - Hopcroft, Tarjan 1971 ; Čadek, Krčál, Matoušek, Sergeraert, Vokřínek, Wagner Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
6 Embed k d d always embeds k never embeds?? Polynomially decidable - Hopcroft, Tarjan 1971 ; Čadek, Krčál, Matoušek, Sergeraert, Vokřínek, Wagner NP-hard - Matoušek, Tancer, Wagner 11 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
7 Embed k d d always embeds k never embeds?? Polynomially decidable - Hopcroft, Tarjan 1971 ; Čadek, Krčál, Matoušek, Sergeraert, Vokřínek, Wagner NP-hard - Matoušek, Tancer, Wagner 11 Undecidable - Matoušek, Tancer, Wagner 11 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
8 Embed k 3 d always embeds k never embeds?? Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
9 Embed k 3 d k D D always embeds never embeds?? Theorem (Matoušek, S, Tancer, Wagner 2014) The following problems are decidable: Embed 2 3, Embed 3 3, and 3-Manifold Embeds in S 3 (or R 3 ). Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
10 Embed k 3 d k D D always embeds never embeds?? Theorem (de Mesmay, Rieck, S, Tancer 2017) The following problems are NP-hard: Embed 2 3, Embed 3 3, and 3-Manifold Embeds in S 3 (or R 3 ). Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
11 Knots and Links Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
12 A link diagram Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
13 Reidemeister moves Reidemeister (1927) Any two diagrams of a link are related by a sequence of 3 moves (shown to the right). Note: Number of crossings may increase before it decreases. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
14 Unlinking Number Crossing Changes: Any link diagram can be made into a diagram of an unlink (trivial) by changing some number of crossings. Unlinking Number: The minimum number of crossings in some diagram that need to be changed to produce an unlink. Warning: Minimum number may not be in the given diagram, so may need Reidemeister moves too. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
15 Unlinking Number Crossing Changes: Any link diagram can be made into a diagram of an unlink (trivial) by changing some number of crossings. Unlinking Number: The minimum number of crossings in some diagram that need to be changed to produce an unlink. Warning: Minimum number may not be in the given diagram, so may need Reidemeister moves too. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
16 Given a link, 3 Questions: Triviality Is it trivial? Can Reidemeister moves produce a diagram with no crossings? Trivial Sub-link Does it have a trivial sub-link? How many components? Unlinking Number What is the unlinking number? How many crossing changes must be made to produce an unlink? Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
17 Hopf link Triviality Doesn t seem trivial, but how do you prove it? Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
18 Linking number for two components: choose red and blue and orient them for crossings of red over blue linking number is the sum of +1 s and 1 s. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
19 Linking number Reidemeister moves don t change the linking number! A crossing change changes the linking number by ±1 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
20 Hopf Link Triviality Not trivial. Linking number is not zero. Trivial Sub-link Maximal trivial sub-link has one component. Unlinking Number Unlinking number 1. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
21 Borromean Rings Triviality Not trivial. (But harder to prove, linking numbers are 0.) Trivial Sub-link Maximal trivial sub-link has two components. Unlinking Number Unlinking number 2. (Must show that it is greater than 1.) Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
22 Borromean Rings Triviality Not trivial. (But harder to prove, linking numbers are 0.) Trivial Sub-link Maximal trivial sub-link has two components. Unlinking Number Unlinking number 2. (Must show that it is greater than 1.) Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
23 Whitehead Double of the Hopf Link Triviality Not trivial. (Requires proof, linking numbers are 0.) Trivial Sub-link Maximal trivial sub-link has one component. Unlinking Number Unlinking number 1. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
24 Whitehead Double of the Borromean Rings Triviality Not trivial. (Requires proof, linking numbers are 0.) Trivial Sub-link Maximal trivial sub-link has two components. Unlinking Number Unlinking number 1. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
25 Decision Problems for Links Triviality Given a link diagram, does it represent a trivial link? (i.e., does it have a diagram with no crossings?) Trivial Sub-link Given a link diagram and a number n, does the link contain a trivial sub-link with n components? Unlinking Number Given a link diagram and a number n, can the link be made trivial by changing n crossings (in some diagram(s))? Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
26 What is known? NP NP-hard Triviality unlikely Trivial Sub-Link Unlinking Number? Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
27 Triviality & Trivial Sub-Link are in NP Haken (1961); Hass, Lagarias, and Pippenger (1999) Unknot recognition is decidable [H], and, in NP [HLP]. Lackenby (2014) For a diagram of an unlink, the number of moves required to eliminate all crossings is bounded polynomially in the number of crossings of starting diagram. Trivial Sub-link is also in NP Apply this to the sub-diagram of the n component trivial sub-link. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
28 Trivial Sub-link is NP-hard Problem: Trivial Sub-link Given a link diagram and a number n, does the link contain a trivial sub-link with n components? Lackenby (2017) (Non-trivial) Sub-link is NP-hard. de Mesmay, Rieck, S and Tancer (2017) Trivial Sub-link is NP-hard Proof is a reduction from 3-SAT: Given an (exact) 3-CNF formula Φ, there is a link L Φ that has an n component trivial sub-link if and only if Φ is satisfiable. (n = number of variables) Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
29 Trivial Sub-link is NP-hard Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
30 Constructing the link L Φ : Φ = (t x y) ( x y z) Given an (exact) 3-CNF formula, need to describe a link. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
31 Constructing the link L Φ : Φ = (t x y) ( x y z) Draw Hopf link for each variable, Borromean rings for each clause. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
32 Constructing the link L Φ : Φ = (t x y) ( x y z) Band each variable to its corresponding variable in the clauses. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
33 Constructing the link L Φ : Φ = (t x y) ( x y z) Band each variable to its corresponding variable in the clauses. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
34 Constructing the link L Φ : Φ = (t x y) ( x y z) Each component is an unknot. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
35 Φ satisfiable = n component trival sub-link Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
36 Satisfiable = n component trivial sub-link : Φ = (t x y) ( x y z) Satisfiable: t = true; x, y, z = false. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
37 Satisfiable = n component trivial sub-link : Φ = (t x y) ( x y z) Erase true components: t, x, y, z. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
38 Satisfiable = n component trivial sub-link : Φ = (t x y) ( x y z) The false components form an n component trivial sub-link. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
39 n component trival sub-link = Φ satisfiable Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
40 n component trivial sub-link = satisfiable: Φ = (t x y) ( x y z) Label the n trivial link components as false, the others true. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
41 n component trivial sub-link = satisfiable: Φ = (t x y) ( x y z) For each pair (x, x), one is true the other false. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
42 n component trivial sub-link = satisfiable: Φ = (t x y) ( x y z) Each clause has a true. (Borromean rings not sub-link of trivial link.) Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
43 n component trivial sub-link = satisfiable: Φ = (t x y) ( x y z) Therefore, Φ is satisfiable. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
44 Unlinking Number is NP-hard Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
45 Unlinking Number is NP-hard Φ = (t x y) ( x y z) Related construction. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
46 Unlinking Number is NP-hard Φ = (t x y) ( x y z) But replace each component with its Whitehead Double! Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
47 Unlinking Number is NP-hard Φ = (t x y) ( x y z) But replace each component with its Whitehead Double! Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
48 Φ satisfiable = unlinking number n Φ = (t x y) ( x y z) Φ is satisfiable, unclasp true components. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
49 Φ satisfiable = unlinking number n Φ = (t x y) ( x y z) The true components are an unlink, push to side. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
50 Φ satisfiable = unlinking number n Φ = (t x y) ( x y z) What remains is also an unlink! = unlinking number n. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
51 unlinking number n = Φ satisfiable Φ = (t x y) ( x y z) Unlinking number n = Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
52 unlinking number n = Φ satisfiable Φ = (t x y) ( x y z) Unlinking number n = each variable gets a crossing change. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
53 unlinking number n = Φ satisfiable Φ = (t x y) ( x y z) Crossing change affects either x or x (not both). Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
54 Φ = (t x y) ( x y z) Call the changed components True Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
55 Φ = (t x y) ( x y z) Every Borromean clause has a changed crossing. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
56 Φ = (t x y) ( x y z) Every Borromean clause has a changed crossing = Φ satisfiable. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
57 Embed 2 3 is NP-hard Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
58 Embed 2 3 is NP-hard : Uses a cabled link and Dehn surgery. Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
59 Open Questions: Knots Links Triviality NP, co-np a NP Trivial Sub-Link n/a NP-complete Unlinking Number? NP-hard 3-Manifold Embeds in S 3 NP b NP-hard Questions: a Kuperberg; Lackenby; b Schleimer 1 Is Unknotting number, i.e., Unlinking Number for a single component, NP-hard? 2 Are Unlinking Number and Embed 2 3 in NP? Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
60 Thanks! Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March / 34
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