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Hard Problems in 3-Manifold Topology School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects Arnaud de Mesmay 1 Yoav Rieck 2 Eric Sedgwick 3 Martin Tancer 4 1 CNRS, GIPSA-Lab 2 University of Arkansas 3 DePaul

  1. Hard Problems in 3-Manifold Topology School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects 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 IHP Paris - June 2018 1 / 40

  2. Some NP -Hard Problems in 3-Manifold Topology Jones Polynomial (# P -hard) - unoriented classical link - Jaeger, Vertigan, Welsh - 1990 Lackenby - 2016 Witten, Reshetikhin, Turaev Heegaard Genus - Bachman, Invariant τ 4 (# P -hard) - Kirby, Derby-Talbot, Sedgwick - 2016 Melvin - 2004 Non Orientable Surface 3-Manifold Knot Genus - Agol, Embeddability - Burton, Hass, Thurston - 2006 de Mesmay, Wagner - 2017 Taut Angle Structure - Burton, Embed 2 → 3 , Embed 3 → 3 , 3-Manifold Embeds in S 3 - Spreer - 2013 de Mesmay, Rieck, Sedgwick, Tancer Turaev-Viro invariants (# P -hard) - - 2017 Burton, Maria, Spreer - 2015 Trivial Sub-Link , Unlinking Immersibility - Burton, Colin de Number , Reidemeister Verdi` ere, de Mesmay - 2016 Distance/Defect, 4-Ball Euler Sublink, Upper Bound for the Char 0 - de Mesmay, Rieck, Thurston complexity of an Sedgwick, Tancer - 2018 Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 2 / 40

  3. Embeddings in R d Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 3 / 40

  4. 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 IHP Paris - June 2018 4 / 40

  5. Does it embed? Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 5 / 40

  6. Does it embed? Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 6 / 40

  7. Does it embed? Yes, but must change the embedding of yellow/green torus from the previous picture. Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 6 / 40

  8. 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 IHP Paris - June 2018 7 / 40

  9. 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 IHP Paris - June 2018 7 / 40

  10. 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 IHP Paris - June 2018 7 / 40

  11. 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 IHP Paris - June 2018 7 / 40

  12. 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 IHP Paris - June 2018 8 / 40

  13. 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 IHP Paris - June 2018 8 / 40

  14. 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 IHP Paris - June 2018 9 / 40

  15. Knots and Links Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 10 / 40

  16. A link diagram Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 11 / 40

  17. Reidemeister moves Reidemeister (1927) Any two diagrams of a link are related by a sequence of 3 moves (shown to the right). Question: Reidemeister Distance How many moves are needed? Note: May need to increase number of crossings. Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 12 / 40

  18. 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 IHP Paris - June 2018 13 / 40

  19. 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 IHP Paris - June 2018 13 / 40

  20. Given a link diagram, 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 IHP Paris - June 2018 14 / 40

  21. Hopf link Triviality Doesn’t seem trivial, but how do you prove it? Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 15 / 40

  22. 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 IHP Paris - June 2018 16 / 40

  23. 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 IHP Paris - June 2018 17 / 40

  24. 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 IHP Paris - June 2018 18 / 40

  25. 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 IHP Paris - June 2018 19 / 40

  26. 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 IHP Paris - June 2018 19 / 40

  27. 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 IHP Paris - June 2018 20 / 40

  28. 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 IHP Paris - June 2018 21 / 40

  29. Reidemeister Distance/Defect Reidemeister Distance Given two diagrams of the same link, let the Reidemeister distance be the number of Reidemeister moves required to get from one to the other. Special Case: Reidemeister Defect Given a diagram of a unlink, how many moves are required to remove all crossings? Measure the defect , the number of extra moves required: Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 22 / 40

  30. Reidemeister Distance/Defect Reidemeister Distance Given two diagrams of the same link, let the Reidemeister distance be the number of Reidemeister moves required to get from one to the other. Special Case: Reidemeister Defect Given a diagram of a unlink, how many moves are required to remove all crossings? Measure the defect , the number of extra moves required: # moves ≥ 1 / 2 crossings Eric Sedgwick (DePaul University) 3-Manifold Topology IHP Paris - June 2018 22 / 40

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