Upsilon-invariants and Alexander polynomials of torus knots Motoo - PowerPoint PPT Presentation

Upsilon-invariants and Alexander polynomials of torus knots Motoo Tange University of Tsukuba 2016/12/20

§ 1. Motivation and results Ozsv´ ath-Stipsicz-Szab´ o defined a concordant invariant Υ K ( t ) (OSS’14/7). • Torus knots • Alternating knots • Linearly independence of concordance group A brief history of Υ after OSS. • Torus knot formula in terms of semigroup (Borodzik and Livingston ’14/8) • Another reasonable definition (Livingston ’15/1) • Υ-invariant of L-space knot and Legendre transform (Borodzik-Hedden ’15/5) • g 4 of some connected-sum of torus knots, (Livingston-Van Cott ’15/8) • L-space knots in terms of formal semigroup (Feller-Krcatovich ’16/2)

• (Infinite) iterated torus knots (not algebraic but L-space) (S.Wang ’16/3) • Whitehead doubles (OSS, Feller-J.Park-Ray ’16/4) • Z ∞ -summand in C ∆ (Kyungbae-M.H.Kim ’16/4) • Inequalities for general cable knots, Non-L-space cable knots (W.Chen ’16/11)

Results Let K be an L-space knot. Υ K p , q ( t ) = ∗ p Υ K ( t ) + Υ T p , q ( t )

Results Let K be an L-space knot. Υ K p , q ( t ) = ∗ p Υ K ( t ) + Υ T p , q ( t ) ∆ K p , q ( t ) = ∆ K ( t p )∆ T p , q ( t ) ( c . f . )

Integration We define integral ∫ 2 I ( K ) = Υ K ( t ) dt 0 This invariant is similar to S 1 -integral of the Tristram-Levine signature.

Integration We define integral ∫ 2 I ( K ) = Υ K ( t ) dt 0 This invariant is similar to S 1 -integral of the Tristram-Levine signature. Torus knot formula ∫ 2 n ∑ Υ T p , q ( t ) dt = − 1 I ( T p , q ) = 3( pq − a i ) 0 i =1 ∫ ( ) S 1 σ ω ( T p , q ) = − 1 pq − 1 p − 1 q + 1 3 pq

§ 2. Definition Definition 1 (L-space) Y : Q HS 3 Y is an L-space ⇔ for s ∈ Spin c ( Y ) HF ( Y , s ) ∼ � = � HF ( S 3 )

§ 2. Definition Definition 1 (L-space) Y : Q HS 3 Y is an L-space ⇔ for s ∈ Spin c ( Y ) HF ( Y , s ) ∼ � = � HF ( S 3 ) Definition 2 (L-space knot) Let K be a knot in S 3 . K is an L-space knot def ⇔ ∃ n ∈ Z > 0 s.t. the n-surgery is an L-space.

§ 2. Definition Definition 1 (L-space) Y : Q HS 3 Y is an L-space ⇔ for s ∈ Spin c ( Y ) HF ( Y , s ) ∼ � = � HF ( S 3 ) Definition 2 (L-space knot) Let K be a knot in S 3 . K is an L-space knot def ⇔ ∃ n ∈ Z > 0 s.t. the n-surgery is an L-space. All L-space knots are fibered knots. (Ni)

§ 2. Definition Definition 1 (L-space) Y : Q HS 3 Y is an L-space ⇔ for s ∈ Spin c ( Y ) HF ( Y , s ) ∼ � = � HF ( S 3 ) Definition 2 (L-space knot) Let K be a knot in S 3 . K is an L-space knot def ⇔ ∃ n ∈ Z > 0 s.t. the n-surgery is an L-space. All L-space knots are fibered knots. (Ni) { Torus knots }⊂{ Algebraic knots } ⊂{ L-space iterated torus knots }⊂{ L-space knots } ⊂{ Strongly quasi-positive knots }⊂{ quasi-positive knot } = { transverse C -link }

Definition 3 (Concordance) Two knots K 0 , K 1 are concordant ⇔ ∃ a smooth annulus embedding f : S 1 × I ֒ → S 3 × I , where I = [0 , 1] and f ( S 1 × i ) = K i , where i = 0 , 1 .

Definition 3 (Concordance) Two knots K 0 , K 1 are concordant ⇔ ∃ a smooth annulus embedding f : S 1 × I ֒ → S 3 × I , where I = [0 , 1] and f ( S 1 × i ) = K i , where i = 0 , 1 . Concordance is an equivalent relation between two knots. { Knots } / ∼ = C sm .

Definition 3 (Concordance) Two knots K 0 , K 1 are concordant ⇔ ∃ a smooth annulus embedding f : S 1 × I ֒ → S 3 × I , where I = [0 , 1] and f ( S 1 × i ) = K i , where i = 0 , 1 . Concordance is an equivalent relation between two knots. { Knots } / ∼ = C sm . Furthermore this set admits group about the connected-sum. This is called the concordance group .

(trefoil) Definition 4 (Knot Floer homology (Ozsv´ ath-Szab´ o)) C ( K ) := CFK ∞ ( K ) A double complex with respect to a Heegaard decomposition of K .

Definition 5 ( Υ -invariant (Ozsv´ ath-Stipsitz-Szab´ o)) � { ( ) } � t 1 − t � ( C ( K ) , F t ) s = x ∈ C ( K ) 2 Alex ( x ) + Alg ( x ) ≤ s 2 ν ( C ( K ) , F t ) = min { s | H 0 (( C ( K ) , F t ) s → H 0 ( C ) = F surj } Υ K ( t ) = − 2 ν ( C ( K ) , F t ) Υ : C → C ([0 , 2]) C ([0 , 2]): the set of continuous functions. Υ K is a piece-wise linear function on [0 , 2].

0.5 1.0 1.5 2.0 - 0.2 - 0.4 - 0.6 - 0.8 - 1.0

Properties(OSS) • Υ : C → C ([0 , 2]) (group homomorphism) 1 Υ K mr = − Υ K 2 Υ K 1 # K 2 = Υ K 1 + Υ K 2 • Υ(2 − t ) = Υ( t ) • Υ ′ K (0) = − τ ( K ) • | Υ K ( t ) | ≤ tg 4 ( K ) (0 < t < 1) • Let K be an alternating. Υ K ( t ) = (1 − | t − 1 | ) σ ( K ) 2 . Definition 6 (Integral of Υ K ) ∫ 2 I : C → R I ( K ) = Υ K ( t ) dt 0 K : an alternating. I ( K ) = σ ( K ) = − τ ( K ) 2

§ 3. Several formulas Fact 7 (Torus knot formula (OSS)) Let K be an L-space knot. ∆ K ( t ) = ∑ n k =0 ( − 1) k t a k (Alexander polynomial) m 0 = 0 , m 2 k = m 2 k − 1 − 1 m 2 k +1 = m 2 k − 2( a 2 k − a 2 k +1 ) + 1 Υ K ( t ) = max 0 ≤ 2 i ≤ n { m 2 i − ta 2 i } a 0 > a 1 > · · · > a 2 n

T (3 , 4) ∆ T (3 , 4) = t 3 − t 2 + 1 − t − 2 + t − 3 m 0 = 0 , m 1 = − 1 , m 2 = − 2 , m 3 = − 5 , m 4 = − 6 a 0 = 3 , a 1 = 2 , a 2 = 0 , a 3 = − 2 , a 4 = − 3

T (3 , 4) ∆ T (3 , 4) = t 3 − t 2 + 1 − t − 2 + t − 3 m 0 = 0 , m 1 = − 1 , m 2 = − 2 , m 3 = − 5 , m 4 = − 6 a 0 = 3 , a 1 = 2 , a 2 = 0 , a 3 = − 2 , a 4 = − 3 0.5 1.0 1.5 2.0 - 1 - 2 - 3 - 4 - 5 - 6

Formal semigroup Fact 8 (Feller-Krcatovich and S.Wang) Let K be an L-space knot. g := g ( K ) Seifert genus { } #( S K ∩ [0 , m )) + t ( g − m ) Υ K ( t ) = − 2 min 2 0 ≤ m ≤ 2 g

Formal semigroup Fact 8 (Feller-Krcatovich and S.Wang) Let K be an L-space knot. g := g ( K ) Seifert genus { } #( S K ∩ [0 , m )) + t ( g − m ) Υ K ( t ) = − 2 min 2 0 ≤ m ≤ 2 g S K :Formal semigroup. 2 n ∑ ( − 1) i t a i ∆ K ( t ) = i =0 (0 = a 0 < a 1 < a 2 < · · · ) ∑ ∞ ∆ K ( t ) 1 − t = t s 0 + t s 1 + t s 2 + · · · = t s n (0 = s 0 < s 1 < s 2 < · · · ) n =0 S K = { s n | n ∈ Z n ≥ 0 } : Formal semigroup

Example( K = T 3 , 7 ) ∆ K ( t ) = 1 − t + t 3 − t 4 + t 6 − t 8 + t 9 − t 11 + t 12 S K = { 0 , 3 , 6 , 7 , 9 , 10 , 12 } ∪ Z n > 12 2 ̸∈ S K ⇔ 11 − 2 ∈ S K 3 ∈ S K ⇔ 11 − 3 ̸∈ S K 4 ̸∈ S K ⇔ 11 − 4 ∈ S K s ∈ S K ⇔ 11 − s ̸∈ S K S K = ⟨ 3 , 7 ⟩ Z ≥ 0 : semigroup generated by 3 , 7. S T p , q = ⟨ p , q ⟩ Z ≥ 0 : semigroup generated by p , q .

Example( K = T 3 , 7 ) ∆ K ( t ) = 1 − t + t 3 − t 4 + t 6 − t 8 + t 9 − t 11 + t 12 S K = { 0 , 3 , 6 , 7 , 9 , 10 , 12 } ∪ Z n > 12 2 ̸∈ S K ⇔ 11 − 2 ∈ S K 3 ∈ S K ⇔ 11 − 3 ̸∈ S K 4 ̸∈ S K ⇔ 11 − 4 ∈ S K s ∈ S K ⇔ 11 − s ̸∈ S K S K = ⟨ 3 , 7 ⟩ Z ≥ 0 : semigroup generated by 3 , 7. S T p , q = ⟨ p , q ⟩ Z ≥ 0 : semigroup generated by p , q . S Pr ( − 2 , 3 , 7) = { 0 , 3 , 5 , 7 , 8 , 10 } ∪ Z n > 10 : not semigroup

Example( K = T 3 , 7 ) ∆ K ( t ) = 1 − t + t 3 − t 4 + t 6 − t 8 + t 9 − t 11 + t 12 S K = { 0 , 3 , 6 , 7 , 9 , 10 , 12 } ∪ Z n > 12 2 ̸∈ S K ⇔ 11 − 2 ∈ S K 3 ∈ S K ⇔ 11 − 3 ̸∈ S K 4 ̸∈ S K ⇔ 11 − 4 ∈ S K s ∈ S K ⇔ 11 − s ̸∈ S K S K = ⟨ 3 , 7 ⟩ Z ≥ 0 : semigroup generated by 3 , 7. S T p , q = ⟨ p , q ⟩ Z ≥ 0 : semigroup generated by p , q . S Pr ( − 2 , 3 , 7) = { 0 , 3 , 5 , 7 , 8 , 10 } ∪ Z n > 10 : not semigroup { } #( S K ∩ [0 , m )) + t ( g − m ) Υ K ( t ) = − 2 min 2 0 ≤ m ≤ 2 g

Formal semigroup of cable knots Formal semigroup S K p , q p ≥ 2 and q ≥ p (2 g ( K ) − 1), then S K p , q = pS K + q Z ≥ 0 . For example: S T (2 , 3) 3 , 5 = 3 ⟨ 2 , 3 ⟩ Z ≥ 0 + 5 Z ≥ 0 = ⟨ 6 , 9 , 5 ⟩ Z ≥ 0

Fact 9 (Torus knot relation (Feller and Krcatovich)) Let p , q be positive integers p , q (with relatively prime). Then, we have Υ T p , q + p = Υ T p , q + Υ T p , p +1

Torus knot formula Let p , q be positive integers as above. 1 q / p = a 1 + = [ a 1 , · · · , a n ] , a 2 + · · · + 1 a n where a i are non-negative integers. Corollary 10 (Continued fraction expansion formula (FK)) n ∑ Υ T p , q = a i Υ p i , p i +1 , i =1 where p i is the denominator of [ a i , · · · , a n ]

Υ( C torus ) = ⟨ Υ p , p +1 | p ∈ Z p ≥ 1 ⟩ C torus : the subgroup generated by torus knots in C . { Υ p , p +1 | p ∈ N > 1 } are linearly independent in C ([0 , 2]).

Corollary 11 (T.) ∑ n I ( T p , q ) = − 1 3( pq − a i ) i =1

Corollary 11 (T.) ∑ n I ( T p , q ) = − 1 3( pq − a i ) i =1 Proof I ( T p i , p i +1 ) = − p 2 i − 1 3 ∑ n ∑ n a i I ( T p i , p i +1 ) = − 1 a i ( p 2 I ( T p , q ) = i − 1) 3 i =1 i =1

From the derivative at t = 0 of Υ T p , q = ∑ n i =1 a i Υ T pi , pi − 1 we have ∑ n ( p − 1)( q − 1) = a i p i ( p i − 1) . (1) i =1 n ∑ a i p i = q + p − 1 (2) i =1 3 ( pq − ∑ n From (1), (2) we have I ( T p , q ) = − 1 i =1 a i ). ✷