USTC-ICTS/PCFT-22-32

Krylov Complexity

in Calabi-Yau Quantum Mechanics

Bao-ning<EMAIL_ADDRESS>Min-xin<EMAIL_ADDRESS>

Interdisciplinary Center for Theoretical Study,

University of Science and Technology of China, Hefei, Anhui 230026, China

Peng Huanwu Center for Fundamental Theory,

Hefei, Anhui 230026, China

Recently, a novel measure for the complexity of operator growth is proposed
based on Lanczos algorithm and Krylov recursion method. We study this Krylov
complexity in quantum mechanical systems derived from some well-known local
toric Calabi-Yau geometries, as well as some non-relativistic models. We find
that for the Calabi-Yau models, the Lanczos coefficients grow slower than
linearly for small $n$’s, consistent with the behavior of integrable models.
On the other hand, for the non-relativistic models, the Lanczos coefficients
initially grow linearly for small $n$’s, then reach a plateau. Although this
looks like the behavior of a chaotic system, it is mostly likely due to
saddle-dominated scrambling effects instead, as argued in the literature. In
our cases, the slopes of linearly growing Lanczos coefficients almost saturate
a bound by the temperature. During our study, we also provide an alternative
general derivation of the bound for the slope.

###### Contents

  1. 1 Introduction
  2. 2 Review of the formalism
  3. 3 A derivation of the bound for the slope of linearly growing Lanczos coefficients
  4. 4 Calabi-Yau models
  5. 5 The quartic anharmonic oscillator
  6. 6 A non-relativistic Toda model
  7. 7 Discussions

##  1 Introduction

The understandings of complexity and chaos are important in both condensed
matter physics and high energy physics, see e.g. [1, 2, 3]. Recently, a novel
measure for the complexity of operator growth is proposed based on Lanczos
algorithm and Krylov recursion method [4], where it is hypothesized that the
asymptotic growth behavior can distinguish between integrable or chaotic
quantum systems. Known as the Krylov complexity, it has been studied
extensively in the literature, e.g. [5, 6, 7, 8, 9, 10]. For studies of other
measures of complexity as a useful indicator of integrable or chaotic nature
of the underlying quantum system, see e.g. the recent papers [11, 12].

In this paper, we compute the Lanczos coefficients and Krylov complexity for
some one-dimensional one-body quantum systems derived from some well-known
local toric Calabi-Yau geometries, studied recently in the context a bootstrap
method in [13]. The Hamiltonians of these models involve exponential functions
of the position and momentum operators, and may provide grounds for exploring
novel phenomena not available in conventional quantum systems. These models
are not only purely of mathematical interest. Indeed, they can describe some
experimentally confirmed phenomena in condensed matter physics such as the
Hofstadter’s butterfly, see e.g. [14, 15]. For comparison purpose, we also
study some non-relativistic models with the standard quadratic kinetic term
for the momentum operator, including the the well-known quartic anharmonic
oscillator model and a Toda model.

These models are expected to be integrable. For examples, for the Calabi-Yau
models, the exact quantization conditions including non-perturbative
contributions were proposed in terms of topological string theory in [16, 17].
The case of local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ Calabi-Yau model also
belongs to the class of integrable systems known as the relativistic Toda
models. We shall test the asymptotic growth hypothesis proposed in [4]. We
find that the Lanczos coefficients of the Calabi-Yau models satisfy $b_{n}\sim
n^{\delta}$, where the scaling exponent $\delta$ is near $\frac{1}{2}$. This
slower than linear growth is consistent with the behavior of integrable
models.

For the quantum chaotic systems, the Lanczos coefficients grow linearly
$b_{n}\sim\alpha n$, and it is argued in [4] that the slope $\alpha$ satisfy a
bound by the temperature

$\alpha\leq\pi T,$ (1.1)

which is related to the famous bound on Lyapunov exponents of chaos in [1]. We
use the convergence property of the autocorrelation function to provide an
alternative probably simpler derivation of the bound.

For the non-relativistic models, we find that the initial growth of $b_{n}$ is
linear. The similar phenomenon also occurs in some other contexts due to the
saddle-dominated scrambling [18, 19], and is not necessarily a decisive
indicator of chaotic behavior. A feature of the non-relativistic models is
that the slopes is proportional to the temperature and the bound (1.1) is
always almost saturated, similar as the inverted harmonic oscillator in [20].

The paper is organized as the followings. In section 2, we review the
formalism of Krylov complexity and use the harmonic oscillator as a warm-up
example. In section 3, we provide a derivation of the bound for the slope of
linearly growing Lanczos coefficients. In section 4, we study the Calabi-Yau
models. In sections 5 and 6, we consider the quartic anharmonic oscillator and
a non-relativistic Toda model respectively. Finally in section 7 we finish
with some discussions.

##  2 Review of the formalism

Consider a quantum system with the Hamiltonian $H$. An operator $\mathcal{O}$
can be regarded as a state in the Hilbert space of operators, denoted as
$|\mathcal{O})$ to distinguish from the bra-ket notation in the usual Hilbert
space. A Wightman inner product, considered e.g. in [5, 21], can be defined on
the Hilbert space of operators

$(\mathcal{O}_{1}|\mathcal{O}_{2})=\frac{\textrm{Tr}(e^{-\frac{\beta
H}{2}}\mathcal{O}_{1}^{\dagger}e^{-\frac{\beta
H}{2}}\mathcal{O}_{2})}{\textrm{Tr}e^{-\beta H}},$ (2.1)

where $\beta=\frac{1}{T}$ is the inverse temperature in our convention. The
definition is properly normalized by the partition function
$Z=\textrm{Tr}e^{-\beta H}$. We can define the norm of an operator
$||\mathcal{O}||:=(\mathcal{O}|\mathcal{O})^{\frac{1}{2}}$ due to the
positivity of inner product.

In the Heisenberg picture of quantum mechanics, the evolution of an operator
can be described by the Liouvillian action

$\mathcal{L}\mathcal{O}:=[H,\mathcal{O}].$ (2.2)

Starting with an operator $|\mathcal{O})$, we can apply the Gram-Schmidt
procedure to orthogonalize the sequence $\\{\mathcal{L}^{n}|\mathcal{O})\\}$.
This is known as the Lanczos algorithm. It is conventional to start with a
Hermitian operator $|\mathcal{O}_{0})$ with proper normalization
$||\mathcal{O}_{0}||=1$. Then we have the next normalized operator
$|\mathcal{O}_{1}):=b_{1}^{-1}\mathcal{L}|\mathcal{O}_{0})$ where
$b_{1}:=||\mathcal{L}\mathcal{O}||$. For $n\geq 2$, the recursion relation is

$\displaystyle|A_{n}):=\mathcal{L}|\mathcal{O}_{n-1})-b_{n-1}|\mathcal{O}_{n-2}),$
(2.3) $\displaystyle
b_{n}:=||A_{n}||,~{}~{}~{}~{}~{}|\mathcal{O}_{n}):=b_{n}^{-1}|A_{n}).$

The recursion may terminate at a finite step if $b_{n}=0$. It is
straightforward to show inductively that the operator $i^{n}\mathcal{O}_{n}$
is Hermitian, so $(\mathcal{O}_{n}|\mathcal{L}|\mathcal{O}_{n})=0$. We have an
orthonormal sequence $|\mathcal{O}_{n}),n=0,1,2,\cdots$ under the Wightman
inner product, known as the Krylov basis. The autocorrelation function is
defined as

$C(t)=(\mathcal{O}_{0}|e^{i\mathcal{L}t}|\mathcal{O}_{0}).$ (2.4)

The sequence of numbers $b_{n}$ is known as the Lanczos coefficients. It is
expected that they initially grow in a ramping region and then eventually
reach a plateau. It is shown that the growth is at most linear [4], and
hypothesized that this is saturated $b_{n}\sim\alpha n$ by chaotic quantum
models, while the growth behavior of integrable models is slower

$b_{n}\sim n^{\delta},~{}~{}~{}~{}0<\delta<1.$ (2.5)

Now we consider the time evolution of an operator
$\mathcal{O}(t)=e^{iHt}\mathcal{O}_{0}e^{-iHt}$ in the Heisenberg picture, and
expand it in the orthonormal Krylov basis as

$|\mathcal{O}(t))=\sum_{n=0}^{\infty}i^{n}\phi_{n}(t)|\mathcal{O}_{n}),$ (2.6)

where the coefficient $\phi_{n}(t)$ can be expressed as

$\phi_{n}(t)=i^{-n}(\mathcal{O}_{n}|\mathcal{O}(t)).$ (2.7)

This means that $\phi_{n}(t)$ can be regarded as the wave function of the
operator, satisfying the normalization condition

$\sum_{n=0}^{\infty}|\phi_{n}(t)|^{2}=1.$ (2.8)

Using the Heisenberg time evolution equation, we can get the differential
equations

$\partial_{t}\phi_{n}=-b_{n+1}\phi_{n+1}+b_{n}\phi_{n-1},~{}~{}~{}\phi_{n}(0)=\delta_{n0},~{}\phi_{-1}(t)=0.$
(2.9)

This equation tell us that all $\phi_{n}(t)(n>1)$ can be solved recursively
with the initial condition $\phi_{0}(t)=C(t)$ which is easily derived from the
Baker-Campbell-Hausdorff formula. Finally, the Krylov complexity is defined as

$K(t):=\sum_{n}n|\phi_{n}(t)|^{2}.$ (2.10)

For the integrable systems, in the early time, it is expected that the Krylov
complexity satisfies polynomial growth:

$K(t)\sim t^{\frac{1}{1-\delta}},$ (2.11)

while for the chaotic systems, it has an exponential growth

$K(t)\sim e^{2\alpha t}.$ (2.12)

The parameters $\delta$ and $\alpha$ are the same as appearing the the growth
of the Lanczos coefficients.

As a warm-up exercise, it is illustrative to work out the Lanczos algorithm
for the simple case of harmonic oscillator

$H=\frac{1}{2}(\hat{x}^{2}+\hat{p}^{2}),$ (2.13)

with the canonical commutation relation $[\hat{x},\hat{p}]=i\hbar$. We can
compute the norm of the operator $\hat{x}$

$\textrm{Tr}(e^{-\frac{\beta H}{2}}\hat{x}e^{-\frac{\beta
H}{2}}\hat{x})=\sum_{m,n=0}^{\infty}e^{-\frac{\beta}{2}(E_{m}+E_{n})}|\langle
m|\hat{x}|n\rangle|^{2},$ (2.14)

where we insert the energy eigenstates and $E_{n}=(n+\frac{1}{2})\hbar$. The
position operator can be written in terms of the creation and annihilation
operators $\hat{x}=\sqrt{\frac{\hbar}{2}}(\hat{a}+\hat{a}^{\dagger})$, so the
sum is only non-vanishing for the cases $m=n\pm 1$ and can be easily done. The
calculations for the momentum operator $\hat{p}$ are similar. Including the
normalization from the partition function $\textrm{Tr}(e^{-\beta
H})=\frac{1}{2\sinh(\frac{\beta\hbar}{2})}$, we find

$||\hat{x}||^{2}=||\hat{p}||^{2}=\frac{\hbar}{2\sinh(\frac{\beta\hbar}{2})}.$
(2.15)

We can apply the Lanczos algorithm starting from a normalized operator

$\mathcal{O}_{0}=\sqrt{\frac{2\sinh(\frac{\beta\hbar}{2})}{\hbar}}\hat{x}.$
(2.16)

The first recursion gives

$\mathcal{O}_{1}=-i\sqrt{\frac{2\sinh(\frac{\beta\hbar}{2})}{\hbar}}\hat{p},~{}~{}~{}b_{1}=\hbar.$
(2.17)

Then the second operator in the recursion (LABEL:recursion) actually vanishes
$A_{2}=0,b_{2}=0$. So the harmonic oscillator is a rather trivial model with
the Lanczos algorithm terminating at the second step.

There is a simple algorithm to compute the Lanczos coefficients from the
moments, explained e.g. in [4]. The moments are defined in terms of the
initial properly normalized operator as

$\mu_{n}:=(\mathcal{O}_{0}|\mathcal{L}^{n}|\mathcal{O}_{0}),$ (2.18)

which is vanishing for odd $n$, since the operator is Hermitian. Then
(correcting a typo in [4]) we have the relation

$b_{1}^{2n}b_{2}^{2(n-1)}\cdots b_{n}^{2}=\det(\mu_{i+j})_{0\leq i,j\leq n}.$
(2.19)

So the Lanczos coefficients can be recursively efficiently computed. For
example, in the above harmonic oscillator example, it is easy to compute
$\mu_{2n}=\hbar^{2n}$ for any $n\geq 0$. Therefore the Lanczos coefficients
are $b_{1}=\hbar,b_{2}=0$.

As a slight generalization, consider a quantum system of $N$ decoupled
harmonic oscillators of different frequencies

$H=\sum_{i=1}^{N}\frac{1}{2}(\hat{p}_{i}^{2}+\omega_{i}^{2}\hat{x}_{i}^{2}),$
(2.20)

with a properly normalized initial operator
$\mathcal{O}_{0}\sim\sum_{i=1}^{N}\hat{x}_{i}$. It is easy to compute the
moments in this case $\mu_{n}=\frac{\hbar^{n}\sum_{i=1}^{N}\omega_{i}^{n}}{N}$
for even $n$. The determinant in (2.19) vanishes for $n\geq 2N$, so the
Lanczos algorithm terminates at the $2N$ step with $b_{2N}=0$.

##  3 A derivation of the bound for the slope of linearly growing Lanczos
coefficients

Before studying concrete models, in this section let us provide another
general discussion relating to the various bounds on quantum chaos. In the
seminal paper [1], a universal bound for the Lyapunov exponent $\lambda_{L}$
of quantum chaos is proposed based on holographic arguments

$\lambda_{L}\leq 2\pi T.$ (3.1)

The Lyapunov exponent can be computed from the four-point OTOC’s (out-of-time-
order correlators), which have been subsequently studied extensively in the
literature, e.g. in a recent paper [22]. On the other hand, for quantum
chaotic systems with linearly growing Lanczos coefficients $b_{n}\sim\alpha
n$, it is proved for infinite temperature and conjectured with strong evidence
for finite temperature in [4] that the Lyapunov exponent is bound by the slope

$\lambda_{L}\leq 2\alpha.$ (3.2)

By analyzing the asymptotic behavior of the spectral function, it is also
argued in [4] that the slope is bound by (1.1). Therefore the bound (3.2) from
Lanczos coefficients is generally stronger than the earlier result (3.1).

Here we provide an alternative derivation of the bound (1.1) without using the
spectral function. Let’s define a new function

$\displaystyle b(\gamma,\epsilon):$
$\displaystyle=\frac{1}{2Z}\\{\textrm{Tr}(e^{(-\frac{\beta}{2}+\gamma)\hat{H}}\mathcal{O}^{\dagger}e^{(-\frac{\beta}{2}+\epsilon)\hat{H}}\mathcal{O})+\textrm{Tr}(e^{(-\frac{\beta}{2}-\gamma)\hat{H}}\mathcal{O}^{\dagger}e^{(-\frac{\beta}{2}-\epsilon)\hat{H}}\mathcal{O})\\}$
(3.3)
$\displaystyle=\frac{1}{Z}\sum_{k=0}^{\infty}\sum_{m+n=2k}\frac{\gamma^{m}\epsilon^{n}}{m!n!}\textrm{Tr}(e^{-\frac{\beta}{2}\hat{H}}\hat{H}^{m}\mathcal{O}^{\dagger}e^{-\frac{\beta}{2}\hat{H}}\hat{H}^{n}\mathcal{O}),$

where $\mathcal{O}=\mathcal{O}_{0}$ is the initial operator in a Lanczos
algorithm.

The moments can also be expressed as

$\displaystyle\mu_{2n}=\frac{1}{Z}\sum_{i+j=2n}\frac{(2n)!}{i!j!}(-1)^{i}\textrm{Tr}(e^{-\frac{\beta}{2}\hat{H}}\hat{H}^{i}\mathcal{O}^{\dagger}e^{-\frac{\beta}{2}\hat{H}}\hat{H}^{j}\mathcal{O}).$
(3.4)

If we set the variable $\epsilon=-\gamma$, we have the following relation

$\displaystyle b(\gamma,-\gamma)$
$\displaystyle=\frac{1}{Z}\textrm{Tr}(e^{(-\frac{\beta}{2}+\gamma)\hat{H}}\mathcal{O}^{\dagger}e^{(-\frac{\beta}{2}-\gamma)\hat{H}}\mathcal{O})$
(3.5) $\displaystyle=\sum_{n=0}^{\infty}\frac{\gamma^{2n}}{(2n)!}\mu_{2n}$
$\displaystyle=(\mathcal{O}|e^{\mathcal{L}\gamma}|\mathcal{O}).$

For $\gamma=0$, $b(0,0)=(\mathcal{O}|\mathcal{O})_{\beta}$ is the norm square
of the operator $\mathcal{O}$, where we denote the temperature parameter
$\beta$ explicitly. Since $b(\gamma,-\gamma)=b(-\gamma,\gamma)$, we can for
convenience assume that $\gamma\geq 0$. And on the third equality of the above
equation, we use the fact that $\mu_{2n+1}=0$. So we find that this is
actually the autocorrelation function $b(\gamma,-\gamma)=C(-i\gamma)$.

We consider the radius of convergence of $b(\gamma,-\gamma)$ as a function of
$\gamma$. We assume that the energy eigenvalues are non-negative and the norm
of square $(\mathcal{O}|\mathcal{O})_{\beta}$ is finite for any finite
temperature $\beta>0$ with the choice of the initial operator $\mathcal{O}$.
Then we can estimate

$\displaystyle b(\gamma,-\gamma)$
$\displaystyle=\frac{1}{Z}\textrm{Tr}(e^{(-\frac{\beta}{2}+\gamma)\hat{H}}\mathcal{O}^{\dagger}e^{(-\frac{\beta}{2}-\gamma)\hat{H}}\mathcal{O})$
(3.6)
$\displaystyle=\frac{1}{Z}\sum_{m,n}e^{(-\frac{\beta}{2}+\gamma)E_{m}+(-\frac{\beta}{2}-\gamma)E_{n}}|\langle
m|\hat{\mathcal{O}}|n\rangle|^{2}$
$\displaystyle<\frac{1}{Z}\sum_{m,n}e^{(-\frac{\beta}{2}+\gamma)E_{m}+(-\frac{\beta}{2}+\gamma)E_{n}}|\langle
m|\hat{\mathcal{O}}|n\rangle|^{2}$
$\displaystyle=\frac{1}{Z}\textrm{Tr}(e^{(-\frac{\beta}{2}+\gamma)\hat{H}}\mathcal{O}^{\dagger}e^{(-\frac{\beta}{2}+\gamma)\hat{H}}\mathcal{O})$
$\displaystyle=(\mathcal{O}|\mathcal{O})_{\beta-2\gamma}.$

So this is finite when $\gamma<\frac{\beta}{2}$, and the radius of convergence
is no less than $\frac{\beta}{2}$. Of course the true radius of convergence
can be much bigger than $\frac{\beta}{2}$. In fact for the case of simple
harmonic oscillator as reviewed in section 2, it is easy to check that
$b(\gamma,-\gamma)$ is finite for any $\gamma$, so the radius of convergence
is infinite in this simple case.

On the other hand, as explained in [4], for the simple example of
$b_{n}=\alpha n$, the autocorrelation function can be computed exactly
$C(t)=\frac{1}{\cosh(\alpha t)}$, while the moments are known as Euler or
secant numbers and have the asymptotic behavior

$\mu_{2n}\sim(\frac{4n\alpha}{e\pi})^{2n}.$ (3.7)

For a general linearly growing Lanczos coefficients $b_{n}\sim\alpha n$ with
some possible small deviations from the exact case, it is expected and checked
in some examples that the above asymptotic behavior is robust and still valid
[4]. Since $\cosh(\alpha t)$ has a zero at $t=\frac{\pi i}{2\alpha}$, the
radius of convergence is $\frac{\pi}{2\alpha}$. Combining the arguments in the
previous paragraph we arrive at the bound (1.1).

##  4 Calabi-Yau models

We consider the local $\mathbb{P}^{1}\times\mathbb{P}^{1}$ model and the local
$\mathbb{P}^{2}$ model, their Hamiltonians are

$\displaystyle e^{\hat{H}}$
$\displaystyle=e^{\hat{x}}+e^{-\hat{x}}+e^{\hat{p}}+e^{-\hat{p}},$ (4.1)
$\displaystyle e^{\hat{H}}$
$\displaystyle=e^{\hat{x}}+e^{\hat{p}}+e^{-\hat{x}-\hat{p}},$

where the position and momentum operators satisfy the canonical commutation
relation $[\hat{x},\hat{p}]=i\hbar$. The energy eigenvalues and eigenstates
can be computed numerically by truncating to a finite level in a basis of
eigenfunctions of a simple harmonic oscillator, as implemented in [23].

The partition function can be expressed as
$Z=\textrm{Tr}e^{-\beta\hat{H}}=\textrm{Tr}\hat{\rho}^{\beta}$, where
$\hat{\rho}:=e^{-\hat{H}}$ is of trace class and positive definite. In some
Calabi-Yau models, there are simple formulas for the operator $\hat{\rho}$ in
the basis of position eigenstates. For the example, the formula for the
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ model is known in the literature, can be
found e.g. in our previous paper [24]

$\hat{\rho}(x_{1},x_{2})=\langle
x_{1}|\hat{\rho}|x_{2}\rangle=\frac{u(x_{1})^{-\frac{1}{2}}u(x_{2})^{-\frac{1}{2}}}{4\hbar\cosh(\frac{\pi(x_{1}-x_{2})}{2\hbar})},$
(4.2)

where $u(x)=2\cosh(\frac{x}{2})$. For the local $\mathbb{P}^{2}$ model, the
formula is more complicated and involves Faddeev’s quantum dilogarithm. An
analytic integral formula the partition function can be obtained when $\beta$
is an integer. In this case, we have

$Z=\textrm{Tr}e^{-\beta H}=\int dx_{1}\int dx_{2}\dots\int
dx_{\beta}\hat{\rho}(x_{1},x_{2})\hat{\rho}(x_{2},x_{3})\dots\hat{\rho}(x_{\beta},x_{1}).$
(4.3)

However the Lanczos algorithm is difficult to implement analytically even for
integer $\beta$, since the Hamiltonians involve logarithm. One possible
approach is to replace the definition of Hamiltonians in (4.1) with
$e^{\hat{H}}\rightarrow\hat{H}$ and correspondingly use a modified Wightman
inner product by replacing the $e^{-\beta\hat{H}}\rightarrow\hat{H}^{-\beta}$
in (2.1). This appears to be a consistent definition since the modified
Wightman inner product would be still positive definite.

Figure 1: The Lanczos coefficients for the Calabi-Yau models.

Figure 2: The Krylov complexity for the Calabi-Yau models.

Here we will not further explore the analytic approach since it can only
possibly work for integer $\beta$ parameters and the integrals still need to
be computed numerically. Instead, we use the numerical approach, which is
rather straightforward with no restriction on parameters. After solving the
energy eigenvalues and eigenstates numerically by truncation in a harmonic
oscillator basis, the moments can be computed by

$\displaystyle\mu_{2n}$
$\displaystyle=(\mathcal{O}_{0}|\mathcal{L}^{n}\mathcal{L}^{n}|\mathcal{O}_{0})$
(4.4) $\displaystyle=\frac{1}{\sum_{i}e^{-\beta
E_{i}}}\sum_{i,j}e^{-\frac{\beta}{2}(E_{i}+E_{j})}(E_{i}-E_{j})^{2n}|\langle
i|\mathcal{O}_{0}|j\rangle|^{2}.$

The calculation of $\mu_{2n}$ is much more straightforward than the norm of
$\mathcal{O}_{n}$ [21, 20]. Then the Lanczos coefficients can be solved by the
relation (2.19).

Computing the Krylov complexity is also straightforward. In our calculations,
we calculate the Lanczos coefficients $b_{n}$’s up to a finite order $n\leq
L$. Then we expand the autocorrelation function $C(t)=b(it,-it)$ as a power
series of $t$ and truncate it at order $2L$. Then the $\phi_{n}$ functions in
(2.7) can be computed recursively as a polynomial function of $t$ of degree
$2L$. It is easy to find that

$\phi_{n}(t)=\frac{(\prod_{i=1}^{n}b_{i})}{n!}t^{n}+c_{n+2}t^{n+2}+\cdots.$
(4.5)

We see that it starts with $t^{n}$ term with a simple formula for the
coefficient and all powers of $t$ have the same parity. The normalization
relation remains consistent with the truncation of the series expansion

$\sum_{i=0}^{L}|\phi_{i}(t)|^{2}=1+O(t^{2L+1}),$ (4.6)

so we can compute the Krylov complexity from (2.10). This should be a good
approximation for the range of small $t$ within the radius of convergence if
the truncation order $L$ is large enough.

We use the truncation method in a basis of harmonic oscillator eigenstates
with the truncation size of $100\times 100$ to numerically solve the Calabi-
Yau models, and set the parameters $\hbar=1$ and $\beta$ in the range of
$1\sim 4$. Note that this truncation size is unrelated to the truncation order
$L$ in the calculations of Krylov complexity. In Figure 1, we choose the
initial operator $\mathcal{O}_{0}=\hat{x}$ and compute the Lanczos
coefficients for the two different Calabi-Yau models. In all cases, the
Lanczos coefficients grow more slowly than linearly before reaching the
plateau, which indicates the Calabi-Yau models are integrable systems. For the
different $\beta$ parameters, the plateaus seem to have the same height. The
regions of growth before reaching plateau are larger as the $\beta$ parameter
increases.

We should note that there are some obvious fluctuations of the Lanczos
coefficients, which is larger for the local
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ model. The fluctuations will be also more
evident in the quartic anharmonic oscillator model in the next section. Such
oscillations of $b_{n}$’s are seen to depend on the parity of $n$, and have
been explained in [26, 27, 28, 20] as due to a small contribution from the
autocorrelation function with the sign $(-1)^{n}$.

In the Figure 2, we plot the Krylov complexity for the Calabi-Yau models. We
use the growth behavior $K(t)\sim t^{\frac{1}{1-\delta}}$ to fit the plots.
The best fit results for the $\delta$ parameter are listed in Table 1.
Interestingly, we see that the parameter is very close to
$\delta\sim\frac{1}{2}$.

Calabi-Yau models | $\beta=1$ | $\beta=2$ | $\beta=4$
---|---|---|---
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ | 0.5031 | 0.4982 | 0.4971
$\mathbb{P}^{2}$ | 0.5004 | 0.4981 | 0.4969
Table 1:  The best fit values of $\delta$ for the Calabi-Yau models.

##  5 The quartic anharmonic oscillator

In this section, we consider the quartic anharmonic oscillator, the
Hamiltonian is given by

$H=p^{2}+x^{2}+gx^{4}.$ (5.1)

The case of a negative quadratic term gives a double-well potential and is
consider recently in [20]. This well known model has been studied in a long
history of literature, see e.g. the early work [25].

There is an extra parameter $g$ comparing to the Calabi-Yau models in the
previous section. In the Calabi-Yau models, the Planck constant $\hbar$
measures the strength of quantum effects and we have made a specific choice
$\hbar=1$. In the current case, we can rescale the position and momentum
operators to set $\hbar=1$, and the strength of quantum effect can be absorbed
into the coupling parameter $g$, while the temperature parameter $\beta$ also
scales accordingly when we perform the Lanczos algorithm. We will perform the
numerical calculations with $\hbar=1$ and $g$ in the range of $\frac{1}{10}$
and $\frac{1}{50}$.

We still choose the initial operator $\mathcal{O}_{0}=\hat{x}$ and truncating
dimension 100. In Figure 3, we give some results about the Lanczos
coefficients from the numerical method. We see that the Lanczos coefficients
grow linearly, and finally reach the plateau. The linear growth may come from
saddle-dominated scrambling and does not necessarily indicate that the model
is chaotic [18, 19]. The slopes in the linearly growing phase is proportional
to $T=\beta^{-1}$ in the range of parameters that we calculate. Furthermore,
in the first figure for a fixed $g=\frac{1}{50}$, we list some results of
ratios of the slope with temperature $\alpha\beta$ in Table 2, and find that
they are less than but very close to $\pi$, almost saturating the bound (1.1).
The same phenomenon also appears in the inverted harmonic oscillator in [20],
and as mentioned there, this is very different from the truly chaotic model
such as the large charge Sachdev-Ye-Kitaev model studied in [4].

In the second figure, we fix $\beta=\frac{1}{4}$ and vary the $g$ parameter.
It shows that the Lanczos coefficients are quite insensitive to $g$ at the
range of $\frac{1}{10}\sim\frac{1}{50}$.

Figure 3: The Lanczos coefficients for the quartic anharmonic oscillator model. Figure 4: The Krylov complexity for the quartic anharmonic oscillator model. $\beta$ | $1$ | $\frac{1}{2}$ | $\frac{1}{4}$
---|---|---|---
$\alpha$ | 3.0896 | 6.1241 | 11.9690
$\alpha\beta$ | 3.0896 | 3.0621 | 2.9922
Table 2:  The number of slope $\alpha$ when $b_{n}$ grows linearly for the
quartic anharmonic oscillator model.

In Figure 4, we plot the $\log(K(t))$ for different $\beta$’s. For different
$g$’s, Lanczos coefficients are almost the same in the range of linear growth,
so the curves of $\log(K(t))$ also almost coincide with each other. We plot
only the case of $g=\frac{1}{50}$ with $\beta=1,\frac{1}{2},\frac{1}{4}$. Like
the paper [20], in our case, the exponential growth is not clearly visible
either.

It is an appropriate place here to discuss some technical details of the
numerical calculations. Generally, the Lanczos coefficients $b_{n}$ is very
difficult to solve analytically and the recursive method will have large error
for as $n$ increases. We solve the Lanczos coefficients from the moments
$\mu_{2n}$’s, which are computed by the equation (4.4). It may seem that for
large energy $E_{i}$, the contribution is suppressed by an exponential factor
so the truncation size does not need to be too large. However if $n$ is large,
the power factor $(E_{i}-E_{j})^{2n}$ may be also very large. So the realistic
truncation size around 100 with our reasonable computational resource may also
produce a large numerical error in this case. Furthermore, as also encountered
in our previous work [13], the frequency of the harmonic oscillator used for
the truncation calculations need to be empirically appropriately chosen, so
that the calculations may converge faster when we increase the truncation
size.

In the previous section, we have noticed that as $\beta$ increases, the region
of growth before reaching plateau is larger, which means we need to compute
more $b_{n}$’s. But when $n$ is large, the calculations of $b_{n}$ will become
more inaccurate as we explained in the above paragraph. On the other hand, if
$\beta$ is small, the larger energy eigenvalues will contribute to the
Wightman inner product, so we have to increase the truncation matrix size in
the basis of eigenfunctions of harmonic oscillator. We choose the parameters
appropriately to balance these competing effects so that we may have more
accurate numerical results with the range of Lanczos coefficients reaching
into the plateau.

##  6 A non-relativistic Toda model

In this section, we consider another class of integrable models, the non-
relativistic Toda models. In particular, we focus on the simplest two-body
case of the Toda chain models, where the center of motion has trivial dynamics
and does not appear in the Hamiltonian. The Hamiltonian for the relative two-
body motion is

$\hat{H}=\frac{p^{2}}{2}+2g\cosh(\hat{x}).$ (6.1)

The exact quantization condition was first derived in [29]. In the modern
approach it is related to the pure $SU(2)$ Seiberg-Witten gauge theory [30,
31].

Similar to the previous sections, here we also choose initial operator
$\mathcal{O}_{0}=\hat{x}$, with $\hbar=1$ and the coupling $g$ in the range of
$\frac{1}{10}$ to $\frac{1}{50}$. In Figure 5, we plot the Lanczos
coefficients, and find that for this integrable model, the figure is similar
to the quartic anharmonic oscillator. At the beginning, the Lanczos
coefficients grow linearly, then they reach the plateau. As shown in the Table
3, they also have the same property that the the slopes are proportional to
$T$, and the ratios are less than but quite close to $\pi$, nearly saturating
the bound (1.1).

In Figure 6, we plot the function $\log(K(t))$ for different $\beta$’s.
Comparing with the quartic anharmonic oscillator model, the Krylov complexity
for this model is closer to exponential growth.

We see that for this model, the properties the Lanczos coefficients and Krylov
complexity are very similar to the quartic anharmonic oscillator in the
previous section. The reason is that for the small coupling $g$, one can
expand the Hamiltonian

$\displaystyle H$ $\displaystyle=p^{2}+g\cosh(x)$ (6.2)
$\displaystyle=g+p^{2}+\frac{g}{2}x^{2}+\frac{g}{24}x^{4}+\dots,$

where the terms of higher powers of $x$ in the potential can be regard as
small perturbations. After some proper rescaling, this can be indeed
transformed into the quartic anharmonic oscillator.

Figure 5: The Lanczos coefficients for the non-relativistic Toda model Figure 6: The Krylov complexity for the non-relativistic Toda model. $\beta$ | $1$ | $\frac{1}{2}$ | $\frac{1}{4}$
---|---|---|---
$\alpha$ | 3.0747 | 6.0157 | 11.4939
$\alpha\beta$ | 3.0747 | 3.0078 | 2.8734
Table 3:  The number of slope $\alpha$ when $b_{n}$ grows linearly for the
non-relativistic Toda model .

##  7 Discussions

In this paper, we study the Lanczos coefficients and Krylov complexity for a
variety of different models using mostly numerical methods. For a linearly
growing Lanczos coefficients, we also provide an alternative derivation on a
bound on the slope by the temperature (1.1) using the radius of convergence in
the autocorrelation function.

For the Calabi-Yau models in section 4, we find that the Lanczos coefficients
grow slower than linear as $b_{n}\sim n^{\delta}$. It is intriguing that the
best fit value of $\delta$ from the Krylov complexity is always quite close to
$\frac{1}{2}$. It would be interesting to search for a universal derivation of
this scaling exponent.

On the other hand, for the non-relativistic models in sections 5 and 6, we
find that the Lanczos coefficients grow linearly at the beginning, and finally
reach the plateau. And for our models, the slopes are proportional to $T$,
with near saturation of the bound (1.1), similar to e.g. the inverted harmonic
oscillator studied in [20]. The linear behavior is due to the saddle-dominated
scrambling effects [18, 19], as we do not expect such simple one-body models
to be truly chaotic. So in these cases the growth of Lanczos coefficients does
not provide a decisive good indicator of the integrable or chaotic nature of
the system. It would be interesting to provide a physical explanation of the
near saturation of the bound (1.1).

In this paper we consider the Krylov complexity defined from operator growth.
It would be interesting to also consider the novel proposal of state
complexity defined by minimizing the spread of a wave function [9]. There are
certainly many other proposed measures of complexity in the literature, e.g.
as recently reviewed in context of the larger topic of quantum information in
holographic duality in [32]. One may also study them in the Calabi-Yau models.

As a first step, in this paper we only consider one-body quantum dynamics. It
would be interesting to couple the quantum systems together and study the
many-body quantum dynamics. It would be certainly interesting to explore the
connections to holographic duality.

Acknowledgments

We thank Jun-hao Li, Gao-fu Ren, Wen-qi Yu, Pei-xuan Zeng for helpful
discussions. This work was supported in parts by the national Natural Science
Foundation of China (Grants No.11947301 and No.12047502).

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# On a potential contact analogue of Kirby move of type 1

Prerak Deep  Department of Mathematics, Indian Institute of Science Education
Research Bhopal<EMAIL_ADDRESS>and  Dheeraj Kulkarni  Department of
Mathematics, Indian Institute of Science Education Research Bhopal
<EMAIL_ADDRESS>
(Date: September 3, 2024)

###### Abstract.

In this note, we explore the possibility of the existence of Kirby move of
type 1 for contact surgery diagrams. In particular, we give the necessary
conditions on a contact surgery diagram to become a potential candidate for
contact Kirby move of type 1. We prove that no other contact integral surgery
diagram satisfies those conditions except for contact $(+2)$-surgery on
Legendrian unknot with Thruston–Bennequin number $-1$.

###### Key words and phrases:

Legendrian knot, Contact surgery, Kirby moves, Contact 3-manifold

###### 2020 Mathematics Subject Classification:

57K33, 53D10, 57R65

##  1\. Introduction

In the 1960s, Lickorish and Wallace showed that any closed connected oriented
3-manifold admits a surgery presentation in terms of a surgery diagram in
$\mathbb{S}^{3}$. In particular, they proved the following result
independently in [11] and [13].

######  Theorem 1 (Lickorish–Wallace).

Any closed oriented connected smooth 3-manifold can be obtained by performing
a sequence of $(\pm 1)$-surgeries on a link in $\mathbb{S}^{3}$.

In 2004, Ding and Geiges proved an analogue result for contact 3-manifolds.
More precisely, they proved the following statement in [2].

######  Theorem 2.

Any closed oriented connected contact 3-manifold can be obtained by a sequence
of contact $(\pm 1)$-surgeries on some Legendrian link in
$(\mathbb{S}^{3},\xi_{st})$.

The above results show the existence of surgery diagrams in the smooth and
contact categories, respectively. However, it is easy to notice that for a
given smooth 3-manifold, there could be more than one surgery diagram
representing it. So, it is natural to ask if there is any relation between the
two surgery diagrams $L_{1}$ and $L_{2}$ corresponding to the same manifold.
Kirby answered this question affirmatively in [10] for the topological
category. Kirby used two types of moves, now known as Kirby moves of type 1
and 2, to obtain $L_{1}$ from $L_{2}$, and vice versa.

In [3], Ding and Geiges discussed the contact analogue of the Kirby move of
type 2. However, they do not discuss the analogue of the Kirby move of type 1.
It is not clear whether one can formulate the analogue of the Kirby move of
type 1. In 2012, R. Avedek in [1] showed that for open books, an analogue of
Kirby’s theorem holds in the contact category. In particular, ribbon moves on
open books play the role of the Kirby moves, as shown in Theorem 1.10 in [1].
Hence, one expects an analogue of the Kirby move of type 1 in terms of contact
surgery diagrams.

In this note, we explore a potential analogue of the Kirby move of type 1 in
the setting of the contact surgery diagram. In particular, we give two
necessary conditions in Observation 1 for a contact surgery diagram to be a
potential contact analogue of the Kirby move of type 1. We classify the
candidate contact integral surgery diagrams into two collections denoted by
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. In Theorem 3, we prove that any
surgery diagram in $\mathcal{C}_{1}$ does not satisfy the necessary
conditions. In Theorem 4, we show that there is a unique surgery diagram in
$\mathcal{C}_{2}$ satisfying the necessary conditions.

##  2\. Preliminaries

Let $M$ be a smooth 3-manifold. A contact structure $\xi$ on $M$ is a
maximally non-integrable hyperplane field, i.e., for a locally defining 1-form
$\alpha$ such that $ker(\alpha)=\xi$ satisfies $\alpha\wedge d\alpha\neq 0$.
Such 1-form $\alpha$ is called a contact form. The pair $(M,\xi)$ is called a
contact manifold. On $\mathbb{S}^{3}\subset\mathbb{R}^{4}$, a standard contact
structure $\xi_{st}$ can be defined as the kernel of the contact 1-form
$\alpha_{st}=\left(x_{1}dy_{1}-y_{1}dx_{1}+x_{2}dy_{2}-y_{2}dx_{2}\right)$.

An embedded circle $K$ in a contact 3-manifold $(M,\xi)$ is called Legendrian
knot if tangential to $\xi$. In this note, we work with Legendrian unknot $K$
in $(\mathbb{S}^{3},\xi_{st})$. A Legendrian unknot is a topologically trivial
Legendrian knot. Due to Eliashberg–Fraser classification in [6], classical
invariants Thurston–Bennequin number and rotation number classify Legendrian
unknots completely. For more details on contact manifolds, Thurston–Bennequin
number and rotation number, the reader can refer to [7].

Dehn surgery is an operation to get a new 3-manifold from a given 3-manifold.
Dehn surgery is performed on a knot $K\subset M$. We cut a thickened
neighbourhood $N(K)$ (homeomorphic to $\mathbb{S}^{1}\times\mathbb{D}^{2}$)
from $M$ and glue a solid torus along the boundary torus via a homeomorphism
$h:\partial(\mathbb{S}^{1}\times\mathbb{D}^{2})\to\partial(N(K))$. After
gluing, we get a 3-manifold,
$N=\overline{M\setminus(N(K))}\cup_{h}\mathbb{S}^{1}\times\mathbb{D}^{2}$. We
call $N$ a surgered manifold obtained from $M$ after performing surgery.

Dehn surgery is completely determined by the image of the meridian $m$,
$h(m)=p\mu+q\lambda$ in $\partial(N(K))$, for a given choice of meridian $\mu$
and longitude $\lambda$ in $\partial(N(K))$. On $N(K)$, $h(m)$ is a curve that
links to $K$ and defines a framing on $K$. In particular, Dehn surgery can be
determined by framed knot $K$ with framing given by a reduced fraction
$\frac{p}{q}$ with $p,q\in\mathbb{Z}$. For example, we get $\mathbb{S}^{3}$
after performing Dehn surgery on $\mathbb{S}^{3}$ along the unknot with
framing $\pm 1$. By Theorem 1, any closed connected oriented smooth 3-manifold
$M$ can be presented by a framed link $L$ (link with framed components) in
$\mathbb{S}^{3}$. Such a framed link $L$ is called a surgery diagram of $M$.
In the surgery diagram, we denote framing on the top of the framed knot $K$.

Let framed links $L_{1}$ and $L_{2}$ correspond to the same closed oriented
connected smooth 3-manifold $M$; then there is a sequence of Kirby moves of
type 1 and 2 to obtain $L_{2}$ from $L_{1}$ by [[10], Theorem 1]. We describe
the Kirby moves as follows. The Kirby move of type 1 is an addition (or
deletion) of unknot with $\pm 1$ framing to (or from) a surgery diagram (see
Figure 1).

Figure 1. Kirby move of type 1

The Kirby move of type 2 is sliding of one component of link $L$ over another.
In particular, given two components $L_{1}$ and $L_{2}$ in ${L}$ with framing
$n_{1}$ and $n_{2}$ respectively, we may slide $L_{1}$ over $L_{2}$ along the
longitude $L^{\prime}_{2}$ determined by the framing. We replace $L_{1}\cup
L_{2}$ by $L^{\prime}_{1}\cup L_{2}$, where
$L^{\prime}_{1}=L_{1}\\#_{b}L^{\prime}_{2}$ (band connected sum of $L_{1}$ and
$L^{\prime}_{2}$) (Figure 2).

Figure 2. Kirby move of type 2

The notion of Dehn surgery has been extended to the contact 3-manifold $M$ in
[2]. We define Dehn surgery on Legendrian knots $K$. We call this surgery a
contact surgery on $M$. Contact surgery on a contact 3-manifold yields a new
contact 3-manifold. Moreover, contact surgery can be shown by a surgery
diagram consisting of a framed Legendrian link (in its front projection). In
contact surgery, the framing curve is determined with respect to the meridian
and the contact longitude, i.e. the longitude curve induced by the contact
structure on $M$.

##  3\. Contact Kirby move of type 1

A reasonable expectation is that a potential contact Kirby move of type 1
would be a Kirby move of type 1 in the topological category. We want to
explore the contact surgery diagram in $(\mathbb{S}^{3},\xi_{st})$, which is
_topologically_ equivalent to Kirby move of type 1.

The Kirby move of type 1 consists of a surgery diagram of $(\pm 1)$-framed
unknot. Thus, any potential contact analogue would consist of a framed
Legendrian unknot with a contact framing corresponding to $(\pm 1)$-framing
for Dehn surgery. Therefore, we explore the relation between the contact
framing on Legendrian unknots and their Dehn surgery framing $\pm 1$. To this
end, we consider the standard tubular neighbourhood $N(K)$, which is
diffeomorphic to $\mathbb{S}^{1}\times\mathbb{D}^{2}$, determined by the
Seifert surface of Legendrian unknot $K$ of $tb(K)=-m$. Recall from [Section
4.1.3, [8]], the standard tubular neighbourhood has coordinates $(z,(x,y))$
and contact 1-form $\alpha=cos(2\pi mz)dy-sin(2\pi mz)dx$ with core
$K=\mathbb{S}^{1}\times{0}=\\{(z,(x,y))\,|\,x=y=0\\}$. On the boundary torus
$\partial N(K)$, we use given Seifert framing to identify $\partial N(K)$ with
$\mathbb{R}^{2}/\mathbb{Z}^{2}$ by taking ${y=0}$ and ${x=0}$ as the meridian
$\mu$ and longitude $\lambda$ curve, respectively. So, the contact longitude
$\lambda_{c}=\lambda-m\mu$.

Let $c$ be the contact framing $n$ curve on boundary torus $\partial N(K)$.
Then $c=\lambda_{c}+n\mu=\lambda+(n-m)\mu$. Notice that $c$ is also a Dehn
surgery framing $\pm 1$ curve on $\partial N(K)$, i.e. $c=\lambda\pm\mu$.
Thus,

(1) $n-m=\pm 1\implies n=m\pm 1.$

We call Equation 1 the topological condition for the existence of a potential
contact Kirby move of type 1.

The Kirby move of type 1 does not change the underlying topological manifold.
It is reasonable to expect its contact analogue to keep the underlying contact
3-manifold unchanged. In other words, adding or deleting a framed Legendrian
unknot from the contact surgery diagram must not change the underlying contact
structure. Therefore, the framed Legendrian unknot $K$ with contact framing
$``n"$ must yield $(\mathbb{S}^{3},\xi_{st})$ back after surgery. Moreover,
the Bennequin inequality forces $tb(K)<0$, which implies that
$0<m\in\mathbb{Z}$. By Equation 1, we get $0\leq n$. We may write these
observations as necessary conditions for a potential contact analogue of Kirby
move of type 1.

######  Observation 1.

The contact surgery diagram, which could be a potential contact Kirby move of
type 1, will satisfy the following necessary conditions.

  1. (1)

It consist of a framed Legendrian unknot $K$ with contact framing
$n\in\mathbb{N}\cup\\{0\\}$ satisfying $n=m\pm 1$, where $tb(K)=-m$.

  2. (2)

After surgery on $K$, we get $(\mathbb{S}^{3},\xi_{st})$ back.

We use Equation 1 to define two collections $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ of contact surgery diagrams satisfying the above conditions.
For $j=1,2$; we define $\mathcal{C}_{j}$ to be the set of contact $n$-surgery
diagrams on framed Legendrian unknot $K$ with $tb(K)=-m$ satisfying above two
conditions, where $n=m+(-1)^{j}$. Now, we state our main theorems.

######  Theorem 3.

There is no contact surgery diagram in $\mathcal{C}_{1}$ satisfying the
necessary conditions of potential contact analogue of Kirby move of type 1.

######  Remark 1.

In $\mathcal{C}_{2}$, Legendrian unknot $K$ with $tb(K)=-1$ with contact
framing $+2$ is a contact surgery diagram satisfying the necessary conditions.
It is also observed by Mark Kegel in Remark 5.5 in [9].

Figure 3. Unique contact integral surgery presentation on unknot for
$(\mathbb{S}^{3},\xi_{st})$.

######  Theorem 4.

In $\mathcal{C}_{2}$, there is no contact surgery diagram satisfying the
necessary conditions of potential contact analogue of Kirby move of type 1
except the unknot $K$ with $tb(K)=-1$ and $rot(K)=0$ shown in Figure 3.

Therefore, a potential contact analogue of Kirby move of type 1 can be stated
as, “Addition (or deletion) of $+2$ contact framed Legendrian unknot with
Thruston–Bennequin number $-1$ to (or from) a given integral contact surgery
diagram.”

##  4\. Proofs of main theorems

In this section, we prove our main theorems. For the proofs, we need to
discuss some terminologies and a lemma.

Suppose $K$ is one of the boundary components of some embedded surface
$\Sigma$ in $\mathbb{S}^{3}$. If $\Sigma$ has more than one boundary
component, then $\Sigma$ is not a Seifert surface of $K$. Let $N(K)$ denote a
standard tubular neighbourhood determined by the Seifert surface of $K$. In
$N(K)$, $\Sigma$ determines a push-off $K^{\prime}$ of $K$ on $\partial N(K)$.
We call $K^{\prime}$ the framing curve induced by $\Sigma$. Now, we can
discuss the following Lemma 1 required for the proofs of the main results.

The more general versions of the following lemma are proved in the literature
(the reader may refer to [Lemma 6.6, [12]], [Theorem 5.4, [9]]). However, we
present a different approach than that of [12, 9].

######  Lemma 1.

Let $K$ be a Legendrian unknot in $(\mathbb{S}^{3},\xi_{st})$ with $tb(K)=-n$
denoting a contact $(n+1)-$surgery (or, contact $(n-1)-$surgery). If $K_{0}$
is a topological framing unknot, then Thurston–Bennequin number of $K_{0}$
changes by $-1$ (or, $+1$) after performing contact surgery on $K$.

###### Proof.

The discussion above shows that a Legendrian $(n\pm 1)-$surgery on a given
Legendrian unknot $K$ is topologically a $(\pm 1)-$surgery. Thus, topological
framing unknot $K_{0}$ has $lk(K,K_{0})=\pm 1$, respectively.

We consider the Hopf link $L=K\cup K_{0}$. Let $S_{L}$ be a Seifert annulus of
$L$. The Seifert annulus $S_{L}$ induces $\pm 1$ framing on $K_{0}$.

Figure 4. Shaded Seifert annulus $S_{L}$ with curve $K^{\prime}$ (in dotted
line) and boundary curves $K\cup K_{0}$.

On the boundary of a small tubular neighbourhood $N(K_{0})$, we denote
$\lambda^{\prime}$ to be a longitude curve determined by the Seifert disk of
$K_{0}$ and closed curve $\mu^{\prime}$ to be a meridian such that
$lk(\lambda^{\prime},\mu^{\prime})=+1$, after fixing orientation on $K_{0}$.
Let $K^{\prime}=S_{L}\cap\partial N(K_{0})$. Since $lk(K_{0},K^{\prime})=\pm
1$, we can express $K^{\prime}=\lambda^{\prime}\pm\mu^{\prime}$.

Suppose $K_{0}$ has $tb(K_{0})=-t$ in $(\mathbb{S}^{3},\xi_{st})$. We may
express contact longitude
$\lambda^{\prime}_{c}=\lambda^{\prime}-t\mu^{\prime}$ in terms of $K^{\prime}$
as below.

(2)
$\lambda^{\prime}_{c}=\lambda^{\prime}-t\mu^{\prime}=(\lambda^{\prime}\pm\mu^{\prime})-(t\pm
1)\mu^{\prime}=K^{\prime}-(t\pm 1)\mu^{\prime}.$

While performing surgery, we remove a thickened neighbourhood $N(K)$ of $K$
such that $\partial(N(K)\cap S_{L})=K^{\prime}$. After removing $N(K)$, we get
a new annulus $S_{L^{\prime}}:=\overline{S_{L}\setminus\left(N(K)\cap
S_{L}\right)}$ with boundary $\partial S_{L^{\prime}}=K^{\prime}\cup K_{0}$.
We attach a meridional disk $D_{m}$ to $K^{\prime}$. We get an embedded disk
$D_{0}=D_{m}\cup S_{L^{\prime}}$ with boundary $K_{0}$ in the new surgered
$\mathbb{S}^{3}$.

In the new $\mathbb{S}^{3}$, $D_{0}$ becomes a Seifert disk of $K_{0}$. It
determines a longitude $\lambda_{new}$ on the boundary of some thickened
neighbourhood $N_{new}(K_{0})\subset N(K_{0})$ of $K_{0}$, and $\lambda_{new}$
is isotopic to $K^{\prime}$. Thus, the contact longitude with respect to the
$\lambda_{new}$ and $\mu^{\prime}$ is $K^{\prime}-(t\pm 1)\mu^{\prime}$ from
Equation 2. Hence $tb_{new}(K_{0})=-(t\pm 1)$ in the new $\mathbb{S}^{3}$. ∎

In the proof of Theorem 3, we show the existence of an overtwisted disk in the
new $\mathbb{S}^{3}$ obtained after surgery on any surgery diagram in
$\mathcal{C}_{1}$. Recall that an embedded disk $D$ in a contact 3-manifold
$(M,\xi)$ is an overtwisted disc, if it has Legendrian boundary whose surface
framing coincides with the contact framing, and in the interior, there is a
point with coinciding contact plane and tangent plane to the disc.

A contact structure $\xi$ on a contact 3-manifold $M$ is called overtwisted if
it contains an overtwisted disc. A contact structure $\xi$ is tight if it is
not an overtwisted contact structure. For example, a standard contact
structure $\xi_{st}$ on $\mathbb{S}^{3}$ is a tight contact structure on
$\mathbb{S}^{3}$. In addition, Eliashberg in [5] proved that $\mathbb{S}^{3}$
has a unique tight contact structure up to isotopy of 2-plane fields, and it
is given by $\xi_{st}$.

###### Proof of Theorem 3.

Suppose $K\in\mathcal{C}_{1}$ with contact framing $n=m-1$, where $tb(K)=-m$.
If $m=1$, it implies a contact 0-surgery on $K$. Contact 0-surgery yields an
overtwisted contact structure on the surgered manifold by definition.

Now we assume $m>1$. Suppose that Legendrian unknot $K$ with $tb(K)=-m$
denotes a contact $(m-1)$-surgery diagram. It corresponds to topological
$(-1)$-surgery on $K$. Suppose $K_{0}$ is a topological framing unknot such
that it is also a Legendrian unknot with $tb(K_{0})=-1$.

In the proof of Lemma 1, after surgery, we obtain a new Seifert disk $D_{0}$
with $\partial D_{0}=K_{0}$ such that $tb_{new}(K_{0})=-(1-1)=0$. In
particular, the contact framing of $K_{0}$ coincides with the surface framing
induced by the new Siefert disk $D_{0}$.

We may perturb $D_{0}$ in the interior such that the contact plane coincides
with the tangent plane at some point. Thus, $D_{0}$ becomes an overtwised disk
in the new surgered ${\mathbb{S}}^{3}$. Hence, the new ${\mathbb{S}}^{3}$ has
an overtwisted contact structure. ∎

In the proof of Theorem 4, we show that there exists a Legendrian unknot $K$
in the new surgered ${\mathbb{S}}^{3}$ violating the Bennequin inequality for
every diagram in $\mathcal{C}_{2}$ except contact $(+2)$-surgery diagram shown
in Figure 3. For any Legendrian unknot $K$ in $(S^{3},\xi_{st})$, the
inequality $tb(K)+|rot(K)|\leq-1$ is known as Bennequin inequality.

###### Proof of Theorem 4.

Suppose unknot $K$, with $tb(K)=-m\neq-1$, is a contact surgery diagram in
$\mathcal{C}_{2}$. The topological surgery framing on $K$ is $+1$, and the
contact framing on $K$ is $m+1$. We draw the topological surgery framing
unknot $K_{0}$ such that it is also a Legendrian unknot $K_{0}$ of
$tb(K_{0})=-1$. Also, note that Bennequin inequality forces $rot(K_{0})=0$.
After contact surgery on $K$, we obtain $\mathbb{S}^{3}$. In the new
$\mathbb{S}^{3}$, the new Thruston–Bennequin number
$tb_{new}(K_{0})=-(1+1)=-2$ by Lemma 1.

We use Lemma 6.6 from [12] to calculate the rotation number of $K_{0}$ in new
$\mathbb{S}^{3}$. Since contact $(m+1)$-surgery on $K$ yields
$(S^{3},\xi_{st})$, we get $K$ to be coarse equivalent (as an unoriented knot)
to Legendrian unknot $U_{m}$ with $tb(U_{m})=-m$ and $rot(U_{m})=|-m+1|$ as in
Figure 5(a) by Theorem 5.4 from [9]. Thus, $K$ is the coarse equivalent (as an
oriented knot) to one of the $U_{m}$ with $rot(U_{m})=-m+1$ or
$rot(U_{m})=m-1$ because $K$ has an orientation.

It is known that in $(\mathbb{S}^{3},\xi_{st})$, the coarse equivalence of two
oriented Legendrian knots is the same as the Legendrian isotopy. Thus, contact
surgeries of the same kind on the coarse equivalent oriented Legendrian knots
produce contactomorphic contact 3-manifolds.

Therefore, we consider two cases:

  1. Case1.

Contact surgery unknot $K$ is coarse equivalent (as an oriented unknot) to
$U_{m}$ such that $rot(U_{m})=-m+1$ and $tb(U_{m})=-m$.

  2. Case2.

Contact surgery unknot $K$ is coarse equivalent (as an oriented unknot) to
$U_{m}$ such that $rot(U_{m})=m-1$ and $tb(U_{m})=-m$.

Notice that the topological surgery framing unknot $K_{0}$ in both cases has
$lk(K,K_{0})=+1$ because topological framing push-off takes the same
orientation as $K$. The calculation below is the same in both cases up to the
rotation number matrix $C$.

First, we convert contact $(m+1)$-surgery on unknot $K$ to a sequence of
contact $(\pm 1)$-surgeries on link $K\cup K_{1}\cup K_{2}\cup\ldots\cup
K_{m}$ using an algorithm described in [4]. In the sequence
$K,K_{1},K_{2},\ldots,K_{m}$, we perform contact $(+1)$-surgery on $K$ and
$(-1)$-surgeries on the rest. Since $\frac{m+1}{-m}$ has a continued fraction
expansion $[-3,-2,\ldots,-2]$ satisfying the condition given in the algorithm,
we add a zigzag to a push-off of $K$ as Figure 5(b) to get $K_{1}$. And
$K_{i}$ is a push-off of $K_{i-1}$, for $2\leq i\leq m$.

(a) $K=U_{m}$

(b) $K\cup K_{1}$

Figure 5. Knot $K$ and its push-off with a zigzag added on the left.

We write the matrix $M$ using the formula given in Lemma 6.6 in [12]. The
entries of $M$ are given by the following

$a_{ij}=\begin{cases}tb(K)+1=-m+1,&\text{ for }i=j=1.\\\
tb(K_{i-1})-1=-m-2,&\text{ for }2\leq i=j\leq m+1.\\\
lk(K,K_{j-1})=tb(K)=-m,&\text{ for }i=1,2\leq j\leq m+1.\\\
lk(K,K_{i-1})=tb(K)=-m,&\text{ for }j=1,2\leq i\leq m+1.\\\
lk\left(K_{i-1},K_{j-1}\right)=tb(K_{i-1})=-m-1,&\text{ for }i\neq j\text{ and
}2\leq i,j\leq m+1.\end{cases}$

Thus,

$M=\left(\begin{array}[]{ccccc}-m+1&-m&-m&\cdots&-m\\\
-m&-m-2&-m-1&\cdots&-m-1\\\ \vdots&\vdots&\vdots&&\vdots\\\
-m&-m-1&-m-1&\cdots&-m-2\end{array}\right)_{(m+1)\times(m+1)}.$

We calculate the inverse

$M^{-1}=\left(\begin{array}[]{ccccc}m^{2}+m+1&-m&-m&\cdots&-m\\\
-m&0&1&\cdots&1\\\ -m&1&0&\cdots&1\\\ \vdots&\vdots&\vdots&&\vdots\\\
-m&1&1&\cdots&0\end{array}\right)_{(m+1)\times(m+1)}.$

Since $K_{1}$ is a push-off of $K$ and $K_{i}$ is a push-off of $K_{i-1}$, for
$2\leq i\leq m$, the linking number $lk(K_{0},K_{j})=+1$ for $1\leq j\leq m$.
Therefore, we get linking number column
$L=\begin{pmatrix}+1&+1&\cdots&+1\end{pmatrix}^{T}.$

Thus, $M^{-1}(L)=\begin{pmatrix}m+1&-1&\cdots&-1\end{pmatrix}^{T}.$

In Case 1, the rotation numbers of $K_{i}$ are the same as $K_{1}$ because
they are push-offs of $K_{i-1}$, for $2\leq i\leq m$. We know that
$rot(K_{1})=rot(K)+1=-m+2$. Thus, the rotation number column
$C=\begin{pmatrix}-m+1&-m+2&\cdots&-m+2\end{pmatrix}^{T}$. Now, we put the
above vectors into the formula below.

$\displaystyle rot_{new}(K_{0})$ $\displaystyle=$ $\displaystyle
rot(K_{0})-\left<C,M^{-1}(L)\right>$ $\displaystyle=$
$\displaystyle-\left<C,M^{-1}(L)\right>\quad(\because rot(K_{0})=0).$
$\displaystyle=$ $\displaystyle-\\{(1-m)(1+m)-m(-m+2)\\}$ $\displaystyle=$
$\displaystyle-(1-m^{2}+m^{2}-2m)$ $\displaystyle=$ $\displaystyle 2m-1.$

Similarly, in Case 2, the rotation number $rot(K_{1})=rot(K)-1=m-2$ and
rotation numbers of $rot(K_{i})=rot(K_{i-1})=rot(K_{1})=m-1$, for $2\leq i\leq
m+1$. Thus, the rotation number column
$C=\begin{pmatrix}m-1&m-2&\cdots&m-2\end{pmatrix}^{T}.$ After putting vectors
in the formula below, we get

$\displaystyle rot_{new}(K_{0})$ $\displaystyle=$
$\displaystyle-\left<C,M^{-1}(L)\right>$ $\displaystyle=$
$\displaystyle-\\{(m-1)(m-1)-m(m-2)\\}$ $\displaystyle=$
$\displaystyle-(-1-m^{2}+m^{2}+2m)$ $\displaystyle=$ $\displaystyle 1-2m.$

Clearly, $|2m-1|>2$ for $m\geq 2$. Thus, in both cases, $K_{0}$ does not
satisfy the Bennequin inequality because $-2+|1-2m|>-1$. Hence, the new
$\mathbb{S}^{3}$ does not have a tight contact structure. ∎

As a consequence, we have proved that there is a unique contact integral
surgery diagram on unknot for $(\mathbb{S}^{3},\xi_{st})$.

######  Corollary 1.

There is a unique contact integral surgery presentation for
$(\mathbb{S}^{3},\xi_{st})$ in $(\mathbb{S}^{3},\xi_{st})$ given by contact
$(+2)$-surgery on Legendrian unknot with Thurston–Bennequin number $-1$.

## Conclusion

In [1], Avdeck shows that ribbon moves are sufficient to relate contact
surgery diagrams corresponding to the same contact 3-manifold. Therefore,
asking whether contact $(+2)$-surgery unknot can be realised as a ribbon move
is natural. We observe that the potential contact Kirby move of type 1 cannot
be converted into a ribbon move as follows. Ribbon moves are defined in [1] as
relations between mapping classes. First, we convert contact $(+2)$-surgery
into a $(\pm 1)-$surgery link $L$ using an algorithm given in [4]. We
construct a Legendrian graph $\overline{L}$ of surgery link $L$ and then the
ribbon surface $R_{\overline{L}}$ of $\overline{L}$ following Algorithm 2 from
[1].

Figure 6. contact $(+2)$-surgery in a sequence of $(\pm 1)-$surgery link $L$
and Legendrian graph $\overline{L}$ of $L$.

In $R_{\overline{L}}$, the surgery link $L$ denotes a mapping class
$D_{L_{1}}^{-1}\circ D_{L_{2}}^{+1}\in MCG(R_{\overline{L}},\partial
R_{\overline{L}})$. But $\psi^{-1}\circ D_{L_{1}}^{-1}\circ
D_{L_{2}}^{+1}\circ\psi\neq Id_{R_{\overline{L}}}$ for any mapping class
$\psi\in MCG(R_{\overline{L}},\partial R_{\overline{L}})$. Hence, the addition
or deletion of contact $(+2)$-surgery unknot does not describe a ribbon move.
Therefore, it remains unclear whether the potential candidate plays the role
of a contact Kirby move of type 1.

## Acknowledgement

This work is part of the PhD Thesis of the first author. The first author is
supported by the Prime Minister Research Fellowship-0400216, Ministry of
Education, Government of India.

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