[NoVIKOV 1962] For n > 5, THERE DOES NOT EXIST AN ALGORITHM WHICH solves: n Computational COMPLEXITY ISSPHERE: Given A TRIANGULATED M IS IT n OF PROBLEMS IN HOMEOMORPHIC TO S ? 3-dimensional TOPOLOGY Thm (Geometrization + MANY Results) TherE IS AN ALGORITHM TO DECIDE IF TWO COMPACT 3-mflDS ARE homeomorphic. Nathan DunfiELD University OF ILLINOIS SLIDES at: http://dunfield.info/preprints/ Today: HoW HARD ARE THESE 3-manifold questions? HoW QUICKLY CAN WE SOLVE them? NP: YES ANSWERS HAVE PROOFS THAT CAN BE Decision Problems: YES OR NO answer. CHECKED IN POLYNOMIAL time. k x F pi(x) = 0 SORTED: Given A LIST OF integers, IS IT sorted? SAT: Given ∈ 2 , CAN CHECK ALL IN LINEAR time. SAT: Given p1,... pn F2[x1,..., xk] IS k ∈ x F pi(x) = 0 i UNKNOTTED: A DIAGRAM OF THE UNKNOT WITH THERE ∈ 2 WITH FOR ALL ? 11 c CROSSINGS CAN BE UNKNOTTED IN O(c ) UNKNOTTED: Given A PLANAR DIAGRAM FOR K 3 Reidemeister MOves. [LackENBY 2013] IN S IS K THE unknot? A Mn(Z) INVERTIBLE: Given ∈ DOES IT HAVE AN INVERSE IN Mn(Z)? coNP: No ANSWERS CAN BE CHECKED IN POLYNOMIAL time. UNKNOTTED: Yes, ASSUMING THE GRH P: Decision PROBLEMS WHICH CAN BE SOLVED [KuperberG 2011]. IN POLYNOMIAL TIME IN THE INPUT size. SORTED: O(LENGTH OF LIST) 3.5 1.1 INVERTIBLE: O n log(LARGEST ENTRY) NKNOTTED IS IN . Conj: U P T KNOTGENUS: Given A TRIANGULATION , A KNOT b1 = 0 (1) IS KNOTGENUS IN P WHEN ? K T g Z 0 K ⊂ , AND A ∈ > , DOES BOUND AN ORIENTABLE SURFACE OF GENUS 6 g? IS THE HOMEOMORPHISM PROBLEM FOR [Agol-Hass-W.Thurston 2006] 3-manifolds IN NP? KNOTGENUS IS NP-complete. What ABOUT DECIDING HYPERBOLICITY? OR Conj (AHT) IF b1(T) = 0, THEN KNOTGENUS BEING AN L-space? IS IN coNP. Computing KhoVANOV HOMOLOGY AND HFK[ [AHT] KNOTAREA IS NP-complete. ARE IN EXPTIME. Just COMPUTING THE Jones POLYNOMIAL IS #P-hard, BUT THE AleXANDER [Dunfield-HirANI 2011] KNOTAREA IS IN P WHEN b1(T) = 0. POLYNOMIAL CAN COMPUTED IN POLY time. Normal SURFACES MEET EACH TETRAHEDRA IN A [Hass-Lagarias-Pippenger 1999] 2 STANDARD way: A FUND SURFACE HAS COORDINATES O exp t . AND CORRESPOND TO LATTICE POINTS IN A fiNITE [AHT 2006] KNOTGENUS IS IN NP. 7t POLYHEDRAL CONE IN R WHERE t = #T: 7t Certificate: A VECTOR x IN Z WITH ENTRIES 2 WITH A MOST O(t ) digits. Check: (1) That x REPRESENTS A NORMAL SURFACE S. (2) That χ(S) 6 1 − 2g. (3) That S CONNECTED AND orientable. (4) That ∂S IS AS advertised. [HakEN 1961] TherE IS A MINIMAL GENUS All CAN BE DONE IN TIME POLYNOMIAL IN t BUT K SURFACE BOUNDING IN NORMAL FORM WHOSE NEED A VERY CLEVER IDEA FOR (3) AND (4). VECTOR IS FUNDAMENTAL (e.g. ON A VERTEX Ray). Hence KNOTGENUS IS decidable. Assuming GRH, [KuperberG 2011] “IN THEORY, THERE IS NO DIffERENCE BETWEEN NKNOTTING IS IN . U coNP THEORY AND PRactice. But, IN PRactice, THERE is.” 3 –Jan L. A. VAN DE Snepscheut ρ: π1(S − K) SL2Fp Certificate: → WHERE log p IS O poly(crossings) . Mystery: IN PRactice, MANY 3-mflD QUESTIONS Check: The FOLLOWING IMPLY π1 IS NOT CYCLIC ARE EASIER THAN THE BEST THEORETICAL BOUNDS AND SO K IS knotted. indicate. (1) Relators FOR π1 hold, SO ρ IS A Rep. Q: HoW BIG A KNOT CAN WE COMPUTE THE GENUS (2) A PAIR OF GENERATORS HAVE for? NONCOMMUTING images. Q: WherE DO WE EVEN GET BIG KNOTS FRom? PrOOF THAT SUCH A REP EXISTS USES ALGEBRAIC TherE ARE MORE 100 CROSSING PRIME KNOTS THAN geometry/number THEORY and: THERE ARE ATOMS IN THE Earth! [Kronheimer-MrOWKA 2004] HerE’S A SNEAK PEAK OF JOINT WORK WITH Malik K S3 Obeidin, BASED ON ONE NATURAL MODEL OF When ⊂ IS nontrivial, THERE IS A REP π (S3 − K) SU RANDOM knot. 1 → 2 WITH NONABELIAN image. Personal best: CRossings: 126 genus: 27 fiBERed: No time: 7 MINUTES HYPERBOLIC volume: 223.6132847441086613 TETRAHEDRa: 243 0 100 80 60 40 20 0 GENUS BOUND FROM AleXANDER POLY −20 −50 0 50 100 150 200 250 300 350 CROSSINGS 105 slope=5.105 104 intercept=-10.037 3 10 r=0.937 102 101 100 alexander_magma_time 10-1 10-2 10-3 101 102 103 crossings 2500 2000 1500 1000 500 HYPERBOLIC VOLUME 0 −500 −200 0 200 400 600 800 1000 1200 CROSSINGS 102 slope=2.331 101 intercept=-6.098 r=0.962 100 10-1 10-2 volume_time 10-3 10-4 10-5 100 101 102 103 104 crossings.