On cobordisms between knots, braid index, and the Upsilon-invariant

Feller, Peter; Krcatovich, David

doi:10.1007/s00208-017-1519-1

Cited by 31 publications

(61 citation statements)

References 33 publications

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“…Proof of Proposition We show that the first singularity

t_{0} > 0

Υ_{K}

is strictly smaller than

\frac{2}{n - 1}

. This suffices since for quasipositive knots (or more generally knots that arise as the closure of braids on which the slice‐Bennequin inequality is sharp) the braid index is bounded below by

\frac{2}{t_{0}}

; see [, Lemma , Proposition 3.7]. Let

g

denote the smooth 4‐ball genus

g_{4} false(K false) = τ false(K false)

K

and let

L

denote the knot obtained as the closure of the

n

‐braid

β_{n, n} = {(δ δ^{normalΔ})}^{n - 1} δ

.…”

Section: Examples and Optimalitymentioning

confidence: 99%

“…The value of

Υ

for torus knots

T_{n, k n + 1}

and

t ⩽ \frac{2}{n}

(see Proposition ) implies the following: For an

n

‐braid, we have

\tilde{Υ} false(t false) = - t \frac{wr (β)}{2}

for

t ⩽ \frac{2}{n}

; see [, Corollary 4.2] or Lemma . Combined with Lemma .III, we can state this as follows.…”

Section: The Homogenization Of Upsilonmentioning

confidence: 99%

“…Let

Δ^{2}

denote the

n

‐braid

{false(a_{1} a_{2} \dots a_{n - 1} false)}^{n}

, called the positive full twist . We have

{\overset{Υ}{true}}_{{normalΔ}^{2} β} = {\overset{Υ}{true}}_{β} + {\overset{Υ}{true}}_{Δ^{2}} = {\overset{Υ}{true}}_{β} + \underset{k \to \infty}{trueprefixlim} \frac{Υ_{T_{n, k n + 1}}}{k} = {\overset{Υ}{true}}_{β} + Υ_{T_{n, n + 1}};

where the first equality holds since

Δ^{2}

is in the center of

B_{n}

(see Lemma II), the second equality follows from choosing

ε_{{({normalΔ}^{2})}^{k}}

to be

a_{1} a_{2} \dots a_{n - 1}

in the definition of

\overset{Υ}{true}

and noting that then the closure of

{({normalΔ}^{2})}^{k} ε_{{false(Δ^{2} false)}^{k}}

is the torus knot

T (n, k n + 1)

, and the third equality is immediate from

Υ_{T_{n, k n + 1}} = k Υ_{T_{n, n + 1}}

(see [, Proposition 2.2]). Using we establish the following.…”

Section: The Fractional Dehn Twist Coefficient As a Slope Of The Homomentioning

confidence: 99%

“…where the first equality holds since Δ 2 is in the center of B n (see Lemma 3.2 II), the second equality follows from choosing ε (Δ 2 ) k to be a 1 a 2 · · · a n−1 in the definition of Υ and noting that then the closure of (Δ 2 ) k ε (Δ 2 ) k is the torus knot T (n, kn + 1), and the third equality is immediate from Υ T n,kn+1 = kΥ Tn,n+1 (see [15,Proposition 2.2]). Using (2) we establish the following.…”

Section: The Fractional Dehn Twist Coefficient As a Slope Of The Homomentioning

confidence: 99%

“…A priori , it does not seem clear that one should even expect

\overset{Υ}{true}

to be linear on

[\frac{2}{n}, \frac{2}{n - 1}]

. In contrast to this, Gambaudo and Ghys studied the homogenization of the

ω

‐signature function, which is also determined by the writhe on

[0, \frac{2}{n}]

: properly normalized it agrees with

\overset{Υ}{true}

[0, \frac{2}{n}]

; see . However, in general the homogenization of the

ω

‐signature function behaves non‐linearly on

[\frac{2}{n}, \frac{2}{n - 1}]

; for example, it is not linear on

[\frac{2}{3}, 1]

for the 3‐braid

a_{1}^{3} a_{2}^{7}

(see [, Example 4.6]).…”

Section: Introductionmentioning

confidence: 99%

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Braids with as many full twists as strands realize the braid index

Feller

Hubbard

2019

Journal of Topology

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We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function‐valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that n‐braids with fractional Dehn twist coefficient larger than n−1 realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that n‐times twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms.

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“…Proof of Proposition We show that the first singularity

t_{0} > 0

Υ_{K}

is strictly smaller than

\frac{2}{n - 1}

. This suffices since for quasipositive knots (or more generally knots that arise as the closure of braids on which the slice‐Bennequin inequality is sharp) the braid index is bounded below by