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Hard Problems in 3-Manifold Topology Einstein Workshop on Discrete Geometry and Topology Arnaud de Mesmay 1 Yoav Rieck 2 Eric Sedgwick 3 Martin Tancer 4 1 CNRS, GIPSA-Lab 2 University of Arkansas 3 DePaul University 4 Charles University Eric

  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 2018 1 / 34

  2. Embeddings in R d Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 2 / 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 2018 3 / 34

  4. Embed k → d d 2 3 4 5 6 7 8 9 10 11 12 13 14 1 always embeds 2 3 k 4 5 never 6 embeds 7 Polynomially decidable - Hopcroft, Tarjan 1971 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 4 / 34

  5. Embed k → d d 2 3 4 5 6 7 8 9 10 11 12 13 14 1 always embeds 2 3 k 4 5 never 6 embeds ? ? 7 Polynomially decidable - Hopcroft, Tarjan 1971 ; ˇ Cadek, Krˇ c´ al, Matouˇ sek, Sergeraert, Vokˇ r´ ınek, Wagner 2013-2017 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 4 / 34

  6. Embed k → d d 2 3 4 5 6 7 8 9 10 11 12 13 14 1 always embeds 2 3 k 4 5 never 6 embeds ? ? 7 Polynomially decidable - Hopcroft, Tarjan 1971 ; ˇ Cadek, Krˇ c´ al, Matouˇ sek, Sergeraert, Vokˇ r´ ınek, Wagner 2013-2017 NP-hard - Matouˇ sek, Tancer, Wagner ’11 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 4 / 34

  7. Embed k → d d 2 3 4 5 6 7 8 9 10 11 12 13 14 1 always embeds 2 3 k 4 5 never 6 embeds ? ? 7 Polynomially decidable - Hopcroft, Tarjan 1971 ; ˇ Cadek, Krˇ c´ al, Matouˇ sek, Sergeraert, Vokˇ r´ ınek, Wagner 2013-2017 NP-hard - Matouˇ sek, Tancer, Wagner ’11 Undecidable - Matouˇ sek, Tancer, Wagner ’11 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 4 / 34

  8. Embed k → 3 d 2 3 4 5 6 7 8 9 10 11 12 13 14 1 always embeds 2 3 k 4 5 never 6 embeds ? ? 7 Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 5 / 34

  9. Embed k → 3 d 2 3 4 5 6 7 8 9 10 11 12 13 14 1 always embeds 2 D D 3 k 4 5 never 6 embeds ? ? 7 Theorem (Matouˇ sek, 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 2018 5 / 34

  10. Embed k → 3 d 2 3 4 5 6 7 8 9 10 11 12 13 14 1 always embeds 2 D D 3 k 4 5 never 6 embeds ? ? 7 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 2018 6 / 34

  11. Knots and Links Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 7 / 34

  12. A link diagram Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 8 / 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 2018 9 / 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 2018 10 / 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 2018 10 / 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 2018 11 / 34

  17. Hopf link Triviality Doesn’t seem trivial, but how do you prove it? Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 12 / 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 2018 13 / 34

  19. Linking number A crossing change Reidemeister moves changes the linking number don’t change the linking by ± 1 number! Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 14 / 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 2018 15 / 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 2018 16 / 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 2018 16 / 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 2018 17 / 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 2018 18 / 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 2018 19 / 34

  26. What is known? NP NP-hard Triviality unlikely � Trivial Sub-Link � � Unlinking Number ? � Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 20 / 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 2018 21 / 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 2018 22 / 34

  29. Trivial Sub-link is NP -hard Eric Sedgwick (DePaul University) 3-Manifold Topology Berlin - March 2018 23 / 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 2018 24 / 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 2018 24 / 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 2018 24 / 34

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