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´

• defined a concordant invariant ΥK(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)

• g4 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. ΥKp,q(t) = ∗pΥK(t) + ΥTp,q(t)

Results

Let K be an L-space knot. ΥKp,q(t) = ∗pΥK(t) + ΥTp,q(t) ∆Kp,q(t) = ∆K(tp)∆Tp,q(t) (c.f .)

Integration

We define integral I(K) = ∫ 2 ΥK(t)dt This invariant is similar to S1-integral of the Tristram-Levine signature.

Integration

We define integral I(K) = ∫ 2 ΥK(t)dt This invariant is similar to S1-integral of the Tristram-Levine signature. Torus knot formula I(Tp,q) = ∫ 2 ΥTp,q(t)dt = −1 3(pq −

n

∑

i=1

ai) ∫

S1 σω(Tp,q) = −1

3 ( pq − 1 p − 1 q + 1 pq )

§2. Definition

Definition 1 (L-space) Y : QHS3 Y is an L-space ⇔ for s ∈ Spinc(Y )

• HF(Y , s) ∼

= HF(S3)

§2. Definition

Definition 1 (L-space) Y : QHS3 Y is an L-space ⇔ for s ∈ Spinc(Y )

• HF(Y , s) ∼

= HF(S3) Definition 2 (L-space knot) Let K be a knot in S3. 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 : QHS3 Y is an L-space ⇔ for s ∈ Spinc(Y )

• HF(Y , s) ∼

= HF(S3) Definition 2 (L-space knot) Let K be a knot in S3. 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 : QHS3 Y is an L-space ⇔ for s ∈ Spinc(Y )

• HF(Y , s) ∼

= HF(S3) Definition 2 (L-space knot) Let K be a knot in S3. 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 K0, K1 are concordant ⇔ ∃ a smooth annulus embedding f : S1 × I ֒ → S3 × I, where I = [0, 1] and f (S1 × i) = Ki, where i = 0, 1.

Definition 3 (Concordance) Two knots K0, K1 are concordant ⇔ ∃ a smooth annulus embedding f : S1 × I ֒ → S3 × I, where I = [0, 1] and f (S1 × i) = Ki, where i = 0, 1. Concordance is an equivalent relation between two knots. {Knots}/ ∼= Csm.

Definition 3 (Concordance) Two knots K0, K1 are concordant ⇔ ∃ a smooth annulus embedding f : S1 × I ֒ → S3 × I, where I = [0, 1] and f (S1 × i) = Ki, where i = 0, 1. Concordance is an equivalent relation between two knots. {Knots}/ ∼= Csm. Furthermore this set admits group about the connected-sum. This is called the concordance group.

Definition 4 (Knot Floer homology (Ozsv´ ath-Szab´

• ))

C(K) := CFK ∞(K)

(trefoil)

A double complex with respect to a Heegaard decomposition of K.

Definition 5 (Υ-invariant (Ozsv´ ath-Stipsitz-Szab´

• ))

(C(K), Ft)s = { x ∈ C(K)

• t

2Alex(x) + ( 1 − t 2 ) Alg(x) ≤ s } ν(C(K), Ft) = min{s|H0((C(K), Ft)s → H0(C) = F surj} ΥK(t) = −2ν(C(K), Ft) Υ : C → C([0, 2]) C([0, 2]): the set of continuous functions. ΥK is a piece-wise linear function on [0, 2].

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Properties(OSS)

• Υ : C → C([0, 2]) (group homomorphism)

1 ΥK mr = −ΥK 2 ΥK1#K2 = ΥK1 + ΥK2

• Υ(2 − t) = Υ(t)
• Υ′

K(0) = −τ(K)

• |ΥK(t)| ≤ tg4(K) (0 < t < 1)
• Let K be an alternating. ΥK(t) = (1 − |t − 1|) σ(K)

2 .

Definition 6 (Integral of ΥK) I : C → R I(K) = ∫ 2 ΥK(t)dt K: an alternating. I(K) = σ(K)

2

= −τ(K)

§3. Several formulas

Fact 7 (Torus knot formula (OSS)) Let K be an L-space knot. ∆K(t) = ∑n

k=0(−1)ktak

(Alexander polynomial) m0 = 0, m2k = m2k−1 − 1 m2k+1 = m2k − 2(a2k − a2k+1) + 1 ΥK(t) = max

0≤2i≤n{m2i − ta2i}

a0 > a1 > · · · > a2n

T(3, 4)

∆T(3,4) = t3 − t2 + 1 − t−2 + t−3 m0 = 0, m1 = −1, m2 = −2, m3 = −5, m4 = −6 a0 = 3, a1 = 2, a2 = 0, a3 = −2, a4 = −3

T(3, 4)

∆T(3,4) = t3 − t2 + 1 − t−2 + t−3 m0 = 0, m1 = −1, m2 = −2, m3 = −5, m4 = −6 a0 = 3, a1 = 2, a2 = 0, a3 = −2, a4 = −3

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Formal semigroup

Fact 8 (Feller-Krcatovich and S.Wang) Let K be an L-space knot. g := g(K) Seifert genus ΥK(t) = −2 min

0≤m≤2g

{ #(SK ∩ [0, m)) + t(g − m) 2 }

Formal semigroup

Fact 8 (Feller-Krcatovich and S.Wang) Let K be an L-space knot. g := g(K) Seifert genus ΥK(t) = −2 min

0≤m≤2g

{ #(SK ∩ [0, m)) + t(g − m) 2 } SK :Formal semigroup. ∆K(t) =

2n

∑

i=0

(−1)itai (0 = a0 < a1 < a2 < · · · ) ∆K(t) 1 − t = ts0 + ts1 + ts2 + · · · =

∞

∑

n=0

tsn (0 = s0 < s1 < s2 < · · · ) SK = {sn|n ∈ Zn≥0}: Formal semigroup

Example(K = T3,7) ∆K(t) = 1 − t + t3 − t4 + t6 − t8 + t9 − t11 + t12 SK = {0, 3, 6, 7, 9, 10, 12} ∪ Zn>12 2 ̸∈ SK ⇔ 11 − 2 ∈ SK 3 ∈ SK ⇔ 11 − 3 ̸∈ SK 4 ̸∈ SK ⇔ 11 − 4 ∈ SK s ∈ SK ⇔ 11 − s ̸∈ SK SK = ⟨3, 7⟩Z≥0: semigroup generated by 3, 7. STp,q = ⟨p, q⟩Z≥0: semigroup generated by p, q.

Example(K = T3,7) ∆K(t) = 1 − t + t3 − t4 + t6 − t8 + t9 − t11 + t12 SK = {0, 3, 6, 7, 9, 10, 12} ∪ Zn>12 2 ̸∈ SK ⇔ 11 − 2 ∈ SK 3 ∈ SK ⇔ 11 − 3 ̸∈ SK 4 ̸∈ SK ⇔ 11 − 4 ∈ SK s ∈ SK ⇔ 11 − s ̸∈ SK SK = ⟨3, 7⟩Z≥0: semigroup generated by 3, 7. STp,q = ⟨p, q⟩Z≥0: semigroup generated by p, q. SPr(−2,3,7) = {0, 3, 5, 7, 8, 10} ∪ Zn>10: not semigroup

Example(K = T3,7) ∆K(t) = 1 − t + t3 − t4 + t6 − t8 + t9 − t11 + t12 SK = {0, 3, 6, 7, 9, 10, 12} ∪ Zn>12 2 ̸∈ SK ⇔ 11 − 2 ∈ SK 3 ∈ SK ⇔ 11 − 3 ̸∈ SK 4 ̸∈ SK ⇔ 11 − 4 ∈ SK s ∈ SK ⇔ 11 − s ̸∈ SK SK = ⟨3, 7⟩Z≥0: semigroup generated by 3, 7. STp,q = ⟨p, q⟩Z≥0: semigroup generated by p, q. SPr(−2,3,7) = {0, 3, 5, 7, 8, 10} ∪ Zn>10: not semigroup ΥK(t) = −2 min

0≤m≤2g

{ #(SK ∩ [0, m)) + t(g − m) 2 }

Formal semigroup of cable knots

Formal semigroup SKp,q p ≥ 2 and q ≥ p(2g(K) − 1), then SKp,q = pSK + qZ≥0. For example: ST(2,3)3,5 = 3⟨2, 3⟩Z≥0 + 5Z≥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 ΥTp,q+p = ΥTp,q + ΥTp,p+1

Torus knot formula

Let p, q be positive integers as above. q/p = a1 + 1 a2 + · · · + 1

an

= [a1, · · · , an], where ai are non-negative integers. Corollary 10 (Continued fraction expansion formula (FK)) ΥTp,q =

n

∑

i=1

aiΥpi,pi+1, where pi is the denominator of [ai, · · · , an]

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

Corollary 11 (T.) I(Tp,q) = −1 3(pq −

n

∑

i=1

ai)

Corollary 11 (T.) I(Tp,q) = −1 3(pq −

n

∑

i=1

ai) Proof I(Tpi,pi+1) = −p2

i − 1

3 I(Tp,q) =

n

∑

i=1

aiI(Tpi,pi+1) = −1 3

n

∑

i=1

ai(p2

i − 1)

From the derivative at t = 0 of ΥTp,q = ∑n

i=1 aiΥTpi ,pi −1 we have

(p − 1)(q − 1) =

n

∑

i=1

aipi(pi − 1). (1)

n

∑

i=1

aipi = q + p − 1 (2) From (1), (2) we have I(Tp,q) = − 1

3(pq − ∑n i=1 ai).

✷

Tristram-Levine signature

σω(K) : S1 → R: Tristram-Levine signature. S: the Seifert matrix σω(K) = sigature((1 − ω)S + (1 − ¯ ω)ST) Fact 12 ∫

S1 σω(Tp,q) = −1

3 ( pq − 1 p − 1 q + 1 pq ) (c.f .) τ(Tp,q) = 1 2(pq − p − q + 1) (c.f .) I(Tp,q) = −1 3 ( pq −

n

∑

i=1

ai )

Cable knot formula

Kp,q : the (p, q)-cable knots, i.e., the satellite knot whose pattern is (p, q)-torus knot in the solid torus. Fact 13 ∆Kp,q = ∆K(tp)∆Tp,q(t), Fact 14 (Tristram-Levine signature formula of cable knots) Let K be a knot. Let p, q be coprime integers. σKp,q(ω) = σK(ωp) + σTp,q(ω)

Fact 15 (τ-invariant formula of Kp,q (Hom)) τ(Kp,q) =      pτ(K) + (p−1)(q−ϵ)

2

ϵ ̸= 0

(p−1)(q−1) 2

ϵ = 0, q > 0

(p−1)(q+1) 2

ϵ = 0, q < 0 In particular, if K is an L-space knot, then τ(Kp,q) = pτ(K) + τ(Tp,q) holds.

L-space cable knots

Fact 16 (Hedden, Hom) Let p, q be positive integers with relatively prime. Kp,q is an L-space knot ⇔ K is an L-space knot and q ≥ p(2g(K) − 1).

§4. Main theorem

Theorem 17 (Upsilon invariant of L-space cable knots (T.)) Let K be an L-space knot. Let p, q be coprime integers with q ≥ 2pg(K). ΥKp,q = ∗pΥK + ΥTp,q. where ∗p is the p-fold juxtaposition of the function.

Corollary 18 (T.) Let K be an L-space knot. Then, I(Kp,q) = I(K) + I(Tp,q). Proof. ∫ 2 ∗pΥKdt = p ∫

2 p

∗pΥK(t)dt = p ∫ 2 ∗pΥK( s p)1 pds = ∫ 2 ΥK(s)ds = I(K) ∫ 2 ΥKp,qdt = ∫ 2 ∗pΥKdt + ∫ 2 ΥTp,qdt ∫ 2 ΥK(t)dt + ∫ 2 ΥTp,qdt = I(K) + I(Tp,q).

Integral values of L-space iterated torus knots

T(p1, q1; p2, q2; · · · ; pn, qn) := (· · · (T(p1, q1)p2,q2)···)pn,qn Theorem 19 (I(K) of L-space iterated torus knots) Let T(p1, q1; p2, q2; · · · ; pn, qn) be an L-space iterated torus knot. I(T(p1, q1; p2, q2; · · · ; pn, qn)) = ∑n

i=1 I(Tpi,qi)

Proof. I(T(p1, q1; p2, q2; · · · ; pn, qn)) = I(T(p1, q1; p2, q2; · · · ; pn−1, qn−1)) + I(Tp1,q1) = · · · = ∑n

i=1 I(Tpi,qi)

Corollary 20 Let K, L be two L-space knots. Let p, q be coprime positive integers with q ≥ 2pg(K), ≥ 2pg(L) ΥKp,q = ΥLp,q ⇔ ΥK = ΥL Conjecture 21 If ΥK(t) = ΥL(t), then ΥKp,q(t) = ΥLp,q(t).

Conjecture 22 (General cabling Upsilon invariant formula) ΥKp,q = ∗pΥK + ΥTp,q.

Conjecture 22 (General cabling Upsilon invariant formula) ΥKp,q = ∗pΥK + ΥTp,q. Counterexamples This conjecture does not true in general. In the case of 2g(K) − 1 ≤ q

p < 2g(K) and Kp,q is an L-space

knot, then ΥKp,q is obtained from the formal semigroup. SKp,q = pSK + qZ≥0.

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Proof. Let K be an L-space. Let SK be a formal semigroup. Ek(t) = { t ≤ k 1 t > k φ(m) = #(SK ∩ [0, m)) = ∑

s∈SK

Es(m)

K = T3,7 SK = {0, 3, 6, 7, 9, 10, 12} ∪ Z>10. SK = {0 = b0, b0+1, · · · , b0+n0−1, b1, b1+1, · · · , b1+n1−1, b2, b2+1, · · · }. Namely, we have ˆ SK = {0 = b0, b1, b2, · · · bk = 2g} ci = bi+1 − (bi + ni). ˆ SK = {0, 3, 6, 9, 12}, k = # ˆ SK = 5. min    ∑

s∈SK

Es(m) − t 2m    = min {

l

∑

i=0

ni − t 2

l−1

∑

i=0

ci|l ∈ 1, 2, · · · , k }

If 2i

p ≤ t < 2(i+1) p

, then we consider the function φ(m) − tm

2

φ(m) − tm 2 = #(SKp,q ∩ [0, m)) − tm 2

There exists a mInimal point around here !

A : ∑

s∈SA Es(m) − tm 2

where SA = {qj mod p|j = 0, 1, 2 · · · , i −1} ⊂ {0, 1, 2, · · · , p −1} B : ∑

s∈SB Es(m) − tm 2

where SB = {qj mod p|j = 0, 1, 2 · · · , i} ⊂ {0, 1, 2, · · · , p − 1}