Direct computation of knot Floer homology and the Upsilon invariant Taketo Sano, joint work with Kouki Sato The University of Tokyo 2019-12-20 1/23
Overview knot homology theory knot concordance invariant knot Floer homology Tau invariant HFK ( K ) τ ( K ) Vk sequence { V k } Upsilon invariant Υ K ( t ) 2/23
I will describe algorithms for computing the G 0 invariant, which is introduced by Kouki Sato. As an application, we obtain algorithms for computing knot concordance invariants such as: 1. τ invariant , a homomorphism from Conc ( S 3 ) to Z , 2. V k sequence , determines the d -invariants of S 3 p / q ( K ) , p / q > 0 , 3. Υ invariant , a homomorphism from Conc ( S 3 ) to PL ([0 , 2] , R ) . Main Result We have determined Υ for almost all (except for 5) knots with crossing number up to 11, including 39 knots whose values have been unknown. ∗ The paper is currently in progress. 3/23
Combinatorial knot Floer homology Knot Floer homology HFK is a knot homology theory, originated from the Heegaard Floer homology. Although HFK involves heavy analytic machineries, a purely combinatorial description was lately found. It uses the grid diagram for the construction, hence also called the grid homology . Figure 1: Grid diagram and the corresponding knot 4/23
Construction of the chain complex (sketch) Let G be a grid diagram of a knot, and N be its grid number. The complex C − ( G ) is a finitely generated free module over F 2 [ U 1 , · · · , U N ], where: ◮ the generators { x } are given by permutations of length N . (Each x can be drawn as N -tuple of points on the lattice), ◮ the homological degree of x is given by the Maslov function M G ( x ), with each factor U i contributing to degree − 2, and ◮ the differential ∂ : C − k ( G ) → C − k − 1 ( G ) is given by � � 1 · · · U ε N U ε 1 ∂ x = N y y r ∈ Rect o ( x , y ) where r runs over the empty rectangles connecting x to y , and the exponents ε 1 , . . . , ε N ∈ { 0 , 1 } are given by counting the number of intersections of r and � ’s. 5/23
� � U ε 1 1 · · · U ε N ∂ x = N y y r ∈ Rect o ( x , y ) r = y = x 6/23
Computing H − ∗ ( G ) algorithmically Since the ground ring F 2 [ U 1 , · · · , U N ] is not a field (nor a PID), we cannot use computational methods to calculate the homology. However, since the generators are finite and deg U i = − 2, we may regard each k -th chain module C − k ( G ) as a finitely generated free F 2 -module with generators of the form: ✿✿✿✿✿✿✿✿✿✿ U a 1 1 · · · U a N deg x − 2 � N x where i a i = k . With these inflated generators , it becomes possible to compute the homology group H − k ( G ) for each k ∈ Z . 7/23
Example ( G = 3 1 , N = 5) k -6 -5 -4 -3 -2 -1 0 # { x } 2 10 27 40 30 10 1 rank C − 622 360 192 90 35 10 1 k rank H − 1 0 1 0 1 0 1 k Example ( G = 6 1 , N = 8) k -11 ... -3 -2 -1 0 1 # { x } 1 ... 8,379 4,949 1,873 402 36 rank C − 5,321,071 ... 24,659 8,165 2,161 402 36 k rank H − 0 ... 0 1 0 1 0 k # of inflated generators explodes as k decreases! 8/23
Proposition ∗ ( G ) ∼ H − = F 2 [ U 1 ] . The homology does not provide any information specific to the knot. 9/23
Bifiltration on C − ∗ ( G ) C − ∗ ( G ) admits a bifiltration . Namely, every inflated generator U a 1 1 · · · U a N N x ∈ C − ∗ ( G ) , is assigned a bidegree ( i , j ) ∈ Z 2 as ◮ i = − a 1 , the exponent of U 1 in its coefficient, and ◮ j = A G ( x ) − � ℓ a ℓ , where A G is the Alexander function . The first degree i is called the algebraic degree , and the second j is called the Alexander degree . It can be proved that ∂ is non-increasing for both i and j . ✿✿✿✿✿✿✿✿✿✿✿✿✿ Important knot concordance invariants such as τ, V k and Υ can be obtained from this bifiltration. In year 2019, K. Sato introduced a new invariant G 0 ( K ) that unifies these invariants. 10/23
Closed region and Z 2 -filtration We call a subset R ⊂ Z 2 an closed region iff: If ( i , j ) ∈ R and ( i ′ ≤ i , j ′ ≤ j ) then ( i ′ , j ′ ) ∈ R . For any closed region R , there is a corresponding subcomplex F R C − := Span F 2 { z ∈ C − | ( i ( z ) , j ( z )) ∈ R } where z is a monomial of the form z = U a 1 1 · · · U a N N x . For another closed region R ′ ⊂ R , we have F R ′ C − ⊂ F R C − . The differential ∂ is closed in F R C − . Thus C − ∗ ( G ) admits a Z 2 -filtration with respect to the partial order ≤ . ✿✿✿✿✿✿✿✿✿✿✿ 11/23
C − j 0 ( G ) R R ′ i 12/23
The invariant G 0 ( K ) = F 2 . The set G (0) 0 ( G ) ∼ Recall that H − 0 ( K ) is defined as the set of minimal closed regions, each containing a homological generator of H − 0 ( G ), namely, � ∃ z ∈ F R C − 0 ( G ) s.t. 0 � = [ z ] ∈ H − 0 ( G ) , R ∈ G (0) 0 ( K ) ⇔ R is minimal w.r.t. the above property. G 0 ( K ) is defined similarly, by regarding homological generators of homological degree ≤ 0. Theorem (Sato ’19, Section 5.1) G 0 ( K ) is a knot concordance invariant. 13/23
Theorem (Sato ’19, Prop. 5.17) G 0 ( K ) determines τ, V k and Υ . j j V k k i i τ Figure 2: Determining τ, V k and Υ from G 0 ( K ). 14/23
Main result Theorem (S.-Sato) There is an algorithm for computing G 0 ( K ) . Corollary There is an algorithm for computing τ, V k and Υ . As an application, we determined Υ for almost all knots of crossing number up to 11, including 39 knots whose Υ have been unknown. The following five are the uncomputed ones, due to computational cost. 10 152 , 11 n 31 , 11 n 47 , 11 n 77 , 11 n 9 . 15/23
Computation of G 0 ( K ) First we compute one homological generator z ∈ C − 0 ( G ), i.e. a 0 ( G ) ∼ cycle whose homology class generates H − = F 2 . For any closed region R , it contains a homological iff: ∃ c ∈ C 1 s . t . z − ∂ c ∈ F R C 0 . Let Q R C ∗ = C ∗ / F R C ∗ , then the above condition is equivalent to: ∃ c ∈ Q R C 1 s . t . ∂ c = z ∈ Q R C 0 Each Q R C i is a finite dimensional F 2 -vector space. We represent ∂ by a matrix A , then the c above corresponds to the solution x of the linear system: Ax = vec ( z ) . 16/23
We minimize the region containing z by “sweeping” its components into a smaller region. The invariant G 0 is the set of all such minimal regions. ∂ � � � � � R ′ R c � � � � z � � � � � z ′ sweep 17/23
Finite candidate regions From Sato’s theorem: − g 4 ( K )[ T 2 , 3 ] ν + ≤ [ K ] ν + ≤ g 4 ( K )[ T 2 , 3 ] ν + , we can tell that for any R ∈ G 0 ( K ), any of its corner ( i , j ) satisfies | i + j | ≤ g 4 ( K ) ( ij ≥ 0) , | i − j | ≤ g 3 ( K ) ( ij < 0) . Thus the set of all closed regions satisfying these conditions is finite, which we can take as a set of candidate regions including G 0 ( K ). 18/23
j g 3 g 4 � � � � � � � � � � � � � � � � � � � i � � � � � � � � � � � � − g 4 − g 3 19/23
The algorithm (sketch) Step 1. Compute one homological generator z ∈ C 0 ( G ). Step 2. Setup the candidate regions. Step 3. Choose one candidate R . Take the basis of Q R C 0 = C 0 / F R C 0 by modding out the generators of C 0 that lie in R . Step 4. Compute the matrix A representing the differential ∂ R : Q R C 1 → Q R C 0 . Step 5. Check whether Ax = vec ( z R ) has a solution. If it does, mark R as realizable. If it doesn’t, discard all regions that are included in R . Goto Step 3 if unchecked candidates exist. Step 6. Collect the realizable R ’s that are minimal w.r.t. the inclusion ⊂ , and we obtain G (0) 0 ( K ). Continue the same process for higher shift numbers. 20/23
DEMO. 21/23
Thank you! 22/23
References I [1] M. Khovanov. “A categorification of the Jones polynomial”. In: Duke Math. J. 101.3 (2000), pp. 359–426. [2] E. S. Lee. “An endomorphism of the Khovanov invariant”. In: Adv. Math. 197.2 (2005), pp. 554–586. [3] J. Rasmussen. “Khovanov homology and the slice genus”. In: Invent. Math. 182.2 (2010), pp. 419–447. [4] C. Manolescu, P. Ozsv´ ath, and S. Sarkar. “A combinatorial description of knot Floer homology”. In: Ann. of Math. (2) 169.2 (2009), pp. 633–660. [5] C. Manolescu et al. “On combinatorial link Floer homology”. In: Geom. Topol. 11 (2007), pp. 2339–2412. K. Sato. The ν + -equivalence classes of genus one knots. 2019. [6] arXiv: 1907.09116 [math.GT] . 23/23
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