# Splitting links by integer homology spheres

Marco Golla  CNRS and Laboratoire des mathématiques Jean Leray, Nantes
Université<EMAIL_ADDRESS>and  Marco Marengon  HUN-REN Alfréd
Rényi Institute of Mathematics<EMAIL_ADDRESS>
###### Abstract.

For every $n\geq 3$, we construct 2-component links in $S^{n+1}$ that are a
split by an integer homology $n$-sphere, but not by $S^{n}$. In the special
case $n=3$, i.e. that of 2-links in $S^{4}$, we produce an infinite family of
links $L_{\ell}$ and of integer homology spheres $Y_{\ell}$ such that the link
$L_{\ell}$ is (topologically or smoothly) split by $Y_{\ell}$ and by no other
integer homology sphere in the family.

##  1\. Introduction

A link in $S^{n+1}$ is the image of an embedding of a disjoint union of
$(n-1)$-spheres, which we consider up to ambient isotopy. We recall that a
link is called (_smoothly_ or _topologically_) _split_ if there is a (smooth
or locally-flat) embedding of $S^{n}$ in $S^{n+1}$ that separates the link,
i.e. each component of $S^{n+1}\setminus S^{n}$ contains at least one
component of the link. More generally, given a closed $n$-manifold $Y$, we say
that a link in $S^{n+1}$ is (_smoothly_ or _topologically_) _split by $Y$_ if
there exists a (smooth or locally-flat) embedding of $Y$ in $S^{n+1}$ that
separates the link.

Moreover, we call links which are (smoothly or topologically) split by an
integer homology sphere (smoothly or topologically) _integrally split_. In
this paper we show that there exist integrally split links which are not split
by the standard $n$-sphere, answering a question raised (in the case $n=3$) by
Peter Kronheimer during Maggie Miller’s talk at the Kronheimer birthday
conference in Oxford.

######  Theorem 1.1.

For every $n\geq 3$, there exists a $2$-component link $L$ in $S^{n+1}$ which
is smoothly integrally split, but not topologically split by $S^{n}$.

We note that in the case $n=2$ (i.e. the case of classical links in $S^{3}$)
the generalised notion of “split by $Y$” is not very useful, because every
link with at least $2$ components is separated by a torus, which is the
simplest possible surface after a 2-sphere. On the other hand, our Theorem 1.1
shows that such a notion is highly non-trivial whenever $n\geq 3$.

We also remark (see Remark 3.2) that there is no real hope to strengthen
Theorem 1.1 by splitting a non-split link by a homotopy sphere: in higher
dimensions, there are no embeddings at all of non-trivial homotopy spheres
into $S^{n+1}$ [Sma62], and in lower dimensions the only non-trivial case
would be when $n=4$, but the existence of non-trivial homotopy $4$-spheres is
still open.

While Theorem 1.1 states the existence of _one_ link for every $n\geq 3$, we
will actually exhibits _infinitely many_ integrally split links that are not
split by $S^{n}$. Moreover, in the case $n=3$ we can produce an infinite
family of links and an infinite family of integer homology $3$-spheres such
that each link in the family is split by exactly one such integer homology
$3$-sphere.

######  Theorem 1.2.

There exist an infinite family $\\{L_{\ell}\\}_{\ell\geq 1}$ of $2$-component
links in $S^{4}$ and an infinite family $\\{Y_{\ell}\\}_{\ell\geq 1}$ of
integer homology $3$-spheres such that $L_{k}$ is smoothly split by $Y_{k}$
but not topologically split by any other $Y_{\ell}$.

The proofs of both Theorems 1.1 and 1.2 is based on a flexible construction of
2-component links $L$ in $S^{n+1}$ which are split by an integer homology
$n$-sphere $Y$ such that the double branched cover $\Sigma_{2}(S^{n+1},L)$ is
a mapping torus on the same $Y$.

In terms of regularity (smooth vs topological), Theorem 1.2 is already the
best possible one (the construction is smooth, the obstruction is
topological). Nonetheless, the topological obstruction from Theorem 1.2 is
quite specific to a certain family of integer homology spheres
$\\{Y_{\ell}\\}_{\ell\in\mathbb{N}}$, so we also prove a smooth obstruction
(using Heegaard Floer homology [OS04b]) which applies more broadly.

######  Theorem 1.3.

Let $L$ be a $2$-component link in $S^{4}$ which is smoothly split by an
integer homology $3$-sphere $Y$, and such that $\Sigma_{2}(S^{4},L)$ is a
mapping torus on $Y$. If $L$ is smoothly split also by an integer homology
$3$-sphere $Z$, then $\operatorname{HF}_{\mathrm{red}}(Y)$ is a direct summand
of $\operatorname{HF}_{\mathrm{red}}(Z)$.

We use Theorem 1.3 to produce another infinite family
$\\{L^{\prime}_{\ell}\\}_{\ell\geq 1}$ of 2-component links in $S^{4}$ and
another infinite family $\\{Y^{\prime}_{\ell}\\}_{\ell\geq 1}$ of
$\mathbb{Z}HS^{3}$ such that $L^{\prime}_{k}$ is smoothly split by
$Y^{\prime}_{k}$ but not by any other $Y^{\prime}_{\ell}$ (see Proposition
5.1).

Ideally we would like to apply the obstruction from Theorem 1.3 to a case when
we know that the link is topologically split, thus detecting exotic splitness.
Therefore, we conclude with the following questions.

######  Question 1.4.

Does there exist a $2$-component link in $S^{4}$ that is topologically but not
smoothly (integrally) split?

######  Question 1.5.

Given an integer homology sphere $Y$, does there exist a $2$-component link in
$S^{4}$ that is topologically but not smoothly split by $Y$?

### Organisation

In Section 2 we present a flexible construction of 2-component links $L$ in
$S^{n+1}$ which are smoothly split by an integer homology sphere $Y$. By our
construction, the double branched cover $\Sigma_{2}(S^{n+1},L)$ is a mapping
torus on $Y$. In Section 3 we show that none of such links are split by
$S^{n}$, and we provide infinitely many explicit examples, hence proving
Theorem 1.1. In Section 4 we prove Theorem 1.2, and in Section 5 we prove the
smooth obstruction (Theorem 1.3). We conclude with an appendix recapping some
well-known properties of Heegaard Floer homology.

### Acknowledgements

We would like to thank Peter Kronheimer for raising the question and Maggie
Miller for the talk on [HKM23] which inspired it. We are also indebted to
Michel Boileau for his suggestions and for pointing us to Rong’s paper
[Ron95]. Finally, we thank Clayton McDonald for his comments on a draft of
this paper. MG thanks the Rényi Institute for their hospitality. MM was
partially supported by NKFIH grant OTKA K146401.

##  2\. The construction

In this section we exhibit a construction of integrally split links in every
dimension. We start with the following proposition.

$S^{n+1}$
---
$S^{n-1}$
---
$S^{n+1}$
---
$S_{K}^{+}$
---
$S_{K}^{-}$
---
$B_{\vartheta}^{n}$
---
$\Sigma_{2}(S^{n},K)\setminus B^{n}$
---
$Y_{K}$
---
$Sp_{1}(K)$
---
$2:1$
---
Figure 1. The open book decomposition of $S^{n+1}$ with pages diffeomorphic to
$B^{n}$ lifts to an open book decomposition of
$\Sigma_{2}(S^{n+1},Sp_{1}(K))$, which is still $S^{n+1}$. The link $L$ is
obtained by taking two translates of the binding $S^{+}_{K}$ and $S_{-}(K)$.

######  Proposition 2.1.

For every knot $K\subset S^{n}$, there is an open book decomposition of
$S^{n+1}$ with binding diffeomorphic to $S^{n-1}$ and pages diffeomorphic to
$\Sigma_{2}(S^{n},K)\setminus B^{n}.$

###### Proof.

Consider the 1-twist spun knot $Sp_{1}(K)\subset S^{n+1}$. By the spin
construction, there is a (trivial) open book decomposition of $S^{n+1}$ with
binding an equatorial $S^{n-1}$ and pages diffeomorphic to $B^{n}_{\vartheta}$
such that for all $\vartheta$ the tangle $Sp_{1}(K)\cap B^{n}_{\vartheta}$ in
$B^{n}_{\vartheta}$ has closure $K$.

By [Zee65, Corollary 2, p. 487], $Sp_{1}(K)$ is unknotted in $S^{n+1}$. Thus,
the double cover $\Sigma_{2}(S^{n+1},Sp_{1}(K))$ of $S^{n+1}$ branched over
$Sp_{1}(K)$ is again $S^{n+1}$. Moreover, we can lift the trivial open book
decomposition to the double branched cover, thus obtaining a new open book
decomposition of $S^{n-1}$. Its page $P_{\vartheta}$ is the double cover of
$B^{n}_{\vartheta}$ branched over (a tangle whose closure is) $K$, i.e.
$\Sigma_{2}(S^{n},K)\setminus B^{n}$; its binding is the double cover of
$S^{n-1}$ branched over an equatorial $S^{n-3}$, hence an $S^{n-1}$. ∎

From such an open book decomposition we deduce the existence of an integrally
split 2-component link, as in the following proposition.

######  Proposition 2.2.

For every knot $K$, there exists a $2$-component link $L_{K}$ in $S^{n+1}$
which is split by the $n$-manifold

$Y_{K}:=\Sigma_{2}(S^{n},K)\\#\Sigma_{2}(S^{n},m(K)),$

and such that $\Sigma_{2}(S^{n+1},L_{K})$ is a mapping torus on $Y_{K}$.

###### Proof.

Consider the open book decomposition of $S^{n+1}$ from Proposition 2.1, and
let $S$ denote its binding. By gluing together two opposite pages
$P_{\vartheta}$ and $P_{-\vartheta}$ along their common boundary $S$, we
obtain an $n$-manifold

$Y_{K}:=\Sigma_{2}(K\\#m(K))=\Sigma_{2}(K)\\#\Sigma_{2}(m(K)),$

which splits $S^{n+1}$ in two diffeomorphic parts. That is, we have exhibited
a splitting of $S^{n+1}$ by $Y_{K}$, i.e.
$S^{n+1}=W^{+}_{K}\cup_{Y_{K}}W^{-}_{K}$. See Figure 1.

We define a link $L_{K}\subset S^{n+1}$ by taking the disjoint union of two
push-offs of the binding $S$, one in each normal direction to $Y_{K}$. Call
these two $(n-1)$-spheres $S^{+}_{K}\subset W^{+}_{K}$ and $S^{-}_{K}\subset
W^{-}_{K}$. Note that in this way, by construction, the link is split by
$Y_{K}$.

We now show that the double cover $p\colon\Sigma_{2}(S^{n+1},L_{K})\to
S^{n+1}$ of $S^{n+1}$ branched over $L_{K}$ is a mapping torus on $Y_{K}$. In
the remaining part of the proof we drop the subscript $K$ for ease of
notation.

Note that $S^{n+1}$ can be viewed as the union of a regular neighbourhood
$N\cong D^{2}\times S$ of $S$ containing $L$ (this is possible since each
component of $L$ is a push-off of $S$) and the exterior $E$ of $S$, which is a
mapping torus on $\Sigma_{2}(S^{n},K)\setminus B^{n}$.

Since each loop in $E$ winding once around $S$ loops twice around $L$, the
restriction of the double cover $p$ to $E$ is the (trivial) disconnected
cover. On the other hand, since $N\cong D^{2}\times S$, the double cover of
$N$ is the product of the identity of $S$ and the double cover of $D^{2}$
branched over two points, so $p^{-1}(N)$ is the product of $S^{n-1}$ and an
annulus.

Thus, $\Sigma_{2}(S^{n+1},L)$ is obtained by gluing the two copies of $E$ to
the two boundary components of $(I\times S^{1})\times S^{n-1}$, in such a way
that the $S^{n-1}$-fibration induced from the mapping torus $E$ on its
boundary $\partial E$ is identified with the standard $S^{n-1}$-fibration of
$S^{1}\times S^{n-1}$, viewed in the boundary of $(I\times S^{1})\times
S^{n-1}$.

It follows that the manifold $\Sigma_{2}(S^{n+1},L)$ is also a mapping torus.
More precisely, since $E$ is the mapping torus of a self-diffeomorphism of
$\Sigma_{2}(S^{n},K)\setminus B^{n}$, it follows that $\Sigma_{2}(S^{n+1},L)$
is the mapping torus of a self-diffeomorphism of $Y$. ∎

From Proposition 2.2 we also obtain an obstruction to the link being split by
another integer homology sphere $Z$.

######  Lemma 2.3.

Let $L$ be a $2$-component link in $S^{n+1}$ which is (smoothly or
topologically) split by an integer homology $n$-sphere $Y$, and such that
$\Sigma_{2}(S^{n+1},L)$ is a mapping torus on $Y$.

If $L$ is (smoothly or topologically) split by another integer homology
$n$-sphere $Z$, then there is a (smooth or topological) invertible cobordism
from $Y$ to $Z$.

Recall that a cobordism $W$ from $Y$ to $Z$ is called _invertible_ if there is
another cobordism $W^{\prime}$ from $Z$ to $Y$ such that the composition
$W^{\prime}\circ W$ is isotopic to $I\times Y$ relative to the boundary. Note
that if $Y$ and $Z$ are integer homology spheres, then both $W$ and
$W^{\prime}$ must necessarily be integer homology cobordisms.

###### Proof.

Suppose that there is a (smooth or topological) embedding of $Z$ in $S^{n+1}$
that splits $L$. If we restrict the branched double cover
$\Sigma_{2}(S^{n+1},L)\to S^{n+1}$ to the complement of $L$, we obtain a
regular Abelian cover. Since $H_{1}(Z;\mathbb{Z})=0$, we can lift the
embedding $Z\hookrightarrow S^{n+1}$ to $\Sigma_{2}(S^{n+1},L)$, which is a
mapping torus on $Y$, and the fact that $Z$ splits $L$ implies that $Z$ lifts
as a homological section of $\Sigma_{2}(S^{n+1},L)$.

The long exact sequence in homotopy associated with the fibration
$Y\to\Sigma_{2}(S^{n+1},L)\to S^{1}$ shows that the image of $\pi_{1}(Y)$
inside $\pi_{1}(\Sigma_{2}(S^{n+1},L))$ is the commutator subgroup, hence the
map

$\mathbb{R}\times Y\to\Sigma_{2}(S^{n+1},L)$

obtained by unwinding the mapping torus is in fact the maximal Abelian cover
of $\Sigma_{2}(S^{n+1},L)$. Again, since $H_{1}(Z;\mathbb{Z})=0$, we can lift
$Z$ to $\mathbb{R}\times Y$, again as a homological section. By choosing two
parallel lifts of $Y$ on either sides of $Z$ we obtain an identity cobordism
$I\times Y$ which is split by $Z$. One of the two components of $(I\times
Y)\setminus Z$ gives a cobordism from $Y$ to $Z$, and the other part is the
left inverse cobordism. ∎

##  3\. Higher-dimensional links

######  Proposition 3.1.

Let $L$ be a $2$-component link in $S^{n+1}$ which is split by an integer
homology $n$-sphere $Y$ with $\pi_{1}(Y)\neq 1$, and such that
$\Sigma_{2}(S^{n+1},L)$ is a mapping torus on $Y$. Then $L$ is not
topologically split by $S^{n}$.

###### Proof.

By Lemma 2.3 there is an invertible topological cobordism from $Y$ to $S^{n}$,
or, equivalently, there is a locally-flat embedding of $S^{n}$ in
$\mathbb{R}\times Y$ as a homological section. Composing with the projection,
this would give a degree-1 map from $S^{n}$ to $Y$. A degree-1 map always
induces an epimorphism on $\pi_{1}$, but this is impossible under the
assumption that $\pi_{1}(Y)\neq 1$. ∎

######  Remark 3.2.

Due to the presence of the hypothesis $\pi_{1}(Y)\neq 1$ in the statement of
Proposition 3.1, the reader may wonder if it is possible to find a 2-component
link in $S^{n+1}$ which is split by an $n$-dimensional _homotopy_ sphere, but
not by $S^{n}$.

When $n\neq 4$ the answer is negative, because it is impossible to even embed
an $n$-dimensional exotic sphere $Y$ in $S^{n+1}$. Indeed, an embedding
$Y\hookrightarrow S^{n+1}$ would give an $h$-cobordism between $Y$ and $S^{n}$
(after removing a small ball from $S^{n+1}$), hence showing that $Y$ is
diffeomorphic to $S^{n}$ [Sma62]. (Note that $n\geq 7$ because $Y$ is assumed
to be exotic.)

When $n=4$ it is unknown whether an exotic $S^{4}$ exists, and the
$h$-cobordism theorem does not hold, so in principle it is possible to split a
link with an exotic $S^{4}$. Doing so would require refuting the smooth
4-dimensional Poincaré conjecture, so we will save this problem for another
day.

######  Theorem 3.3.

For every $n\geq 3$, there exists a $2$-component link $L$ in $S^{n+1}$ which
is smoothly split by an integer homology $n$-sphere $Y$ but is not
topologically split by $S^{n}$.

###### Proof.

Let $J$ be a non-trivial knot in $S^{3}$ with $\det J=1$, so that
$\Sigma_{2}(S^{3},J)$ is an integer homology sphere with
$\pi_{1}(\Sigma_{2}(S^{3},J))\neq 1$. (Concretely, we can choose an infinite
family of such knots by taking $J$ to be any torus knot $T(p,q)$ with $p,q>1$
both odd, in which case $\Sigma_{2}(S^{3},J)$ is the Brieskorn sphere
$\Sigma(2,p,q)$.)

We define a knot $K$ in $S^{n}$ by taking the $0$-spun knot $n-3$ times:

$K:=Sp_{0}\left(\cdots\left(Sp_{0}(J)\right)\cdots\right).$

From the $0$-spinning construction, it is straightforward to check that if $J$
is a knot whose double branched cover is an integer homology sphere, then so
is $Sp_{0}(J)$. Thus, iterating this reasoning, we have that
$\Sigma_{2}(S^{n},K)$ is an integer homology sphere. Moreover, since
$0$-spinning preserves the knot group, we have that $\pi_{1}(S^{n}\setminus
K)\cong\pi_{1}(S^{3}\setminus J)$, and therefore
$\pi_{1}(\Sigma_{2}(S^{n},K))\cong\pi_{1}(\Sigma_{2}(S^{3},J))$, being their
unique index-$2$ subgroups: thus, $\Sigma_{2}(S^{n},K)$ is also an integer
homology sphere with $\pi_{1}(\Sigma_{2}(S^{n},K))\neq 1$.

By Proposition 2.2, there exists a link $L_{K}$ in $S^{n+1}$ which is split by
the manifold $Y_{K}=\Sigma_{2}(S^{n},K)\\#\Sigma_{2}(S^{n},m(K))$ (which by
the previous paragraph is an integer homology sphere with $\pi_{1}(Y_{K})\neq
1$), and such that $\Sigma_{2}(S^{n+1},L)$ is a mapping torus over $Y_{K}$. It
then follows from Proposition 3.1 that $L_{K}$ is not topologically split by
$S^{n}$. ∎

##  4\. Locally-flatly splitting in dimension $4$

In this section we prove the following more precise version of Theorem 1.2.

######  Proposition 4.1.

For a positive torus knot $T=T(p,q)\neq T(3,5)$ with $p,q$ both odd, let
$L_{T}$ and $Y_{T}$ denote the link and the integer homology spheres given by
Proposition 2.2. If $T,T^{\prime}$ are two positive torus knots as above, then
$L_{T}$ is smoothly split by $Y_{T}$ but not topologically split by
$Y_{T^{\prime}}$.

We are indebted to Michel Boileau for suggesting the idea of the proof.

Recall that if $T=T(p,q)$ is a torus knot with $p$, $q$ both odd, then $\det
T=1$ and the double cover $\Sigma_{2}(S^{3},T)$ is the Brieskorn sphere
$\Sigma(2,p,q)$, which is a small Seifert fibred space. If, furthermore,
$T\neq T(3,5)$, then $\pi_{1}(\Sigma(2,p,q))$ is infinite. Since small Seifert
fibred spaces are irreducible, by the sphere theorem the universal cover of
$\Sigma(2,p,q)$ has trivial $\pi_{2}$. Since the fundamental group of
$\Sigma(2,p,q)$ is infinite, the universal cover has trivial $H_{3}$, and by
the Hurewicz theorem $\Sigma(2,p,q)$ has trivial $\pi_{3}$ too.

Moreover, there are no incompressible surfaces in $\Sigma(2,p,q)$, and the
only connected incompressible surface in
$\Sigma(2,p,q)\\#\overline{\Sigma(2,p,q)}$ is the separating 2-sphere. It
follows that the only incompressible surfaces in
$\Sigma(2,p,q)\\#\overline{\Sigma(2,p,q)}$ are parallel copies of the
separating 2-sphere. Indeed, any incompressible surface lives in one of the
summands (by an innermost circle argument on the separating 2-sphere).

As the reader might have anticipated, we will be interested in looking at
degree-$1$ maps between connected sums of Brieskorn spheres. Rong has studied
degree-1 maps between Seifert manifolds in [Ron95] and proved such maps are
not so common. In particular, [Ron95, Theorem 3.2] asserts that if there
exists a degree-$1$ map between two Brieskorn spheres $\Sigma(2,p,q)$ and
$\Sigma(2,p^{\prime},q^{\prime})$ with infinite fundamental group (with either
orientation), then111This is easy to verify using the condition on the least
common multiples of the multiplicities of the fibres in the definition of the
partial order. $\\{p,q\\}=\\{p^{\prime},q^{\prime}\\}$.

###### Proof.

Suppose that there are $T=T(p,q)$ and $T^{\prime}=(p^{\prime},q^{\prime})$
positive torus knots with $p,q,p^{\prime},q^{\prime}$ all odd such that
$L_{T}$ is split by $Y_{T^{\prime}}$. For convenience, call
$Y=\Sigma(2,p^{\prime},q^{\prime})$ and $Z=\Sigma(2,p,q)$, so that
$Y_{T^{\prime}}\cong Y\\#\overline{Y}$ and $Y_{T}\cong Z\\#\overline{Z}$.

By Lemma 2.3 there is an invertible cobordism from $Z\\#\overline{Z}$ to
$Y\\#\overline{Y}$, and therefore a topological embedding of
$Y\\#\overline{Y}$ in $\mathbb{R}\times(Z\\#\overline{Z})$ such that post-
composing with the projection $\mathbb{R}\times(Z\\#\overline{Z})\to
Z\\#\overline{Z}$ yields a continuous degree-$1$ map

$f\colon Y\\#\overline{Y}\to Z\\#\overline{Z}.$

We want to show that such an $f$ cannot exist. Up to perturbations, we can
suppose that $f$ is smooth.

Call $S$ the sphere that splits $Z\\#\overline{Z}$ into $Z\setminus B^{3}$ and
$\overline{Z}\setminus B^{3}$. Up to a small perturbation, we can suppose that
$f$ is transverse to $S$, and that in particular $f^{-1}(S)$ is a surface in
$Y\\#\overline{Y}$. Up to performing some embedded surgeries, as explained in
[Lau74, Section I.2.3], we can suppose that $f^{-1}(S)$ is a union of
incompressible surfaces.

Since $Y$ is a small Seifert fibred space, the only connected incompressible
surface in $Y\\#\overline{Y}$ is the separating sphere, so $f^{-1}(S)$ is a
union of parallel spheres. Suppose that there are at least two spheres in
$f^{-1}(S)$, and choose two consecutive ones, which cobound a 3-manifold $A$,
with $A\cong S^{2}\times[0,1]$. The restriction $f_{A}$ of $f$ to $A$ maps $A$
either into $Z\setminus B^{3}$ or into $\overline{Z}\setminus B^{3}$ (by
connectedness). Suppose that we are in the former case (the latter being
entirely equivalent). Note that $f(\partial A)\subset\partial(Z\setminus
B^{3})$, so we can extend $f_{A}$ to a map $g_{A}\colon S^{3}\to Z$, where we
view $S^{3}$ as obtained by filling the boundary of $A$ by two 3-balls.

Since $Z$ is a Seifert fibred space with infinite fundamental group,
$\pi_{3}(Z)=0$, and therefore $g_{A}$ is null-homotopic, and in particular it
has degree $0$. It follows that also $f_{A}$ has degree $0$ (since $A$ is the
preimage of $Z\setminus B^{3}\subset Z$, and we can compute the degree by
counting the points in the preimage of a generic point).

Since the map $f$ has degree $1$, and its restriction to all these copies of
$S^{2}\times[0,1]$ has degree $0$, the preimage of $Z\setminus B^{3}$ cannot
be contained in a union of these copies, and in particular it must contain,
say, $Y\setminus B^{3}\subset Y\\#\overline{Y}$. It follows that the
restriction to $Y\setminus B^{3}$ has degree $1$ onto $Z\setminus B^{3}$. We
can now extend the restriction of $f$ to $Y\setminus B^{3}$ to a degree-$1$
map $g\colon Y\to Z$, but no such map exists by work of Rong [Ron95, Theorem
3.2], as explained above.

This concludes the proof. ∎

##  5\. Smoothly splitting in dimension $4$

Since the proof of Proposition 4.1 uses strong geometric assumptions on the
manifolds $Y_{K}$, we also prove a smooth obstruction, namely Theorem 1.3 from
the introduction. In Theorem 1.3 we use the Heegaard Floer homology package.
We recall the notation and the properties we will need in Appendix A.

###### Proof of Theorem 1.3.

Lemma 2.3 implies that there is an invertible cobordism $W$ from $Y$ to $Z$,
with inverse $W^{\prime}$. Since $W$ and $W^{\prime}$ are integer homology
cobordisms, the cobordism maps $F_{W}$ and $F_{W^{\prime}}$ in
$\operatorname{HF}_{\mathrm{red}}$ preserve the Maslov grading.

By functoriality, the composition

$\operatorname{HF}_{\mathrm{red}}(Y)\xrightarrow{F_{W}}\operatorname{HF}_{\mathrm{red}}(Z)\xrightarrow{F_{W^{\prime}}}\operatorname{HF}_{\mathrm{red}}(Y)$

is the identity map, so we have a splitting of graded $\mathbb{F}[U]$-modules

$\operatorname{HF}_{\mathrm{red}}(Z)\cong\ker
F_{W^{\prime}}\oplus\operatorname{im}F_{W},$

and $\operatorname{im}F_{W}\cong\operatorname{HF}_{\mathrm{red}}(Y)$, as
desired.

This argument has appeared for invertible concordances in [JM16]. A similar
argument for ribbon concordances and cobordisms received much attention in
last years, sparked by [Zem19] (see [DLVW22] for the case of ribbon homology
cobordisms). We were also inspired by [LLP23]. ∎

We apply Theorem 1.3 to produce a family of examples (in the smooth category)
which is not covered by Proposition 4.1.

######  Proposition 5.1.

For each $\ell\geq 2$, let $K_{\ell}=T(4\ell+1,8\ell+3)\\#T(3,5)$, and let
$L_{K_{\ell}}$ and $Y_{K_{\ell}}$ denote the links and the integer homology
spheres given by Proposition 2.2.

For each $m,n\geq 2$, the link $L_{K_{n}}$ in $S^{4}$ is smoothly split by
$Y_{K_{n}}$ but not by $Y_{K_{m}}$ for any $m\neq n$.

The following definition is a compact way of referring to integer homology
spheres with $\operatorname{HF}_{\rm red}$ of a special form, which is
convenient for a set of examples that we will study. We say that a homology
3-sphere $Y$ is _of size_ $\ell\geq 2$ if:

  1. (1)

the length of the longest finite tower in
$\operatorname{HF}^{-}_{\mathrm{even}}(Y)$ is $\ell$;

  2. (2)

there exists at least one finite tower of length $1$ in positive degree in
$\operatorname{HF}^{-}_{\mathrm{even}}(Y)$, and the smallest positive degree
that supports a length-1 tower is $2(\ell-1)$.

Note that not all homology 3-spheres have a well-defined size.

######  Remark 5.2.

If $Y_{m}$ and $Y_{n}$ are integer homology spheres of different sizes $m$ and
$n$ respectively, then neither $\operatorname{HF}_{\rm red}(Y_{m})$ nor
$\operatorname{HF}_{\rm red}(Y_{n})$ is a direct summand of the other. Suppose
without loss of generality that $m>n$: then property (1) obstructs
$\operatorname{HF}_{\rm red}(Y_{m})$ from being a direct summand of
$\operatorname{HF}_{\rm red}(Y_{n})$, while property (2) obstructs the other
inclusion.

######  Example 5.3.

Call $\Sigma_{\ell}=\Sigma(2,4\ell+1,8\ell+3)$. We want to show that
$Y_{\ell}=\Sigma_{\ell}\\#\overline{\Sigma}_{\ell}$ has size $\ell$ for each
$\ell\geq 2$.

Rustamov computed $\operatorname{HF}^{+}(\overline{\Sigma}_{\ell})$ for every
$\ell$ [Rus03, Theorem 1.1], and from his computation we can immediately
deduce $\operatorname{HF}^{-}(\Sigma_{\ell})$ using [OS04a, Proposition 2.5].
Specifically, we have:

  * •

$d(\Sigma_{\ell})=0$;

  * •

$\operatorname{HF}^{-}(\Sigma_{\ell})$ is entirely supported in non-positive
even degree;

  * •

there are exactly two towers of length $1$ in
$\operatorname{HF}^{-}(\Sigma_{\ell})$, supported in Maslov gradings
$-\ell(\ell+1)$ and $-(\ell-1)\ell$;

  * •

there is exactly one tower of length $\ell$ in
$\operatorname{HF}^{-}(\Sigma_{\ell})$, which is the maximal possible length
among finite towers;

  * •

the closest grading supporting the generator of a finite tower in
$\operatorname{HF}^{-}(\Sigma_{\ell})$ to $-\ell(\ell+1)$ or $-(\ell-1)\ell$
is $-(\ell-2)(\ell-1)$.

Now, both properties (1) and (2) above for
$Y_{\ell}=\Sigma_{\ell}\\#\overline{\Sigma}_{\ell}$ follow from [OS04a,
Corollary 6.3] and [OS04a, Proposition 2.5].

Indeed, since $\operatorname{HF}^{-}_{\rm red}(\Sigma_{\ell})$ is supported in
even degree, we have that $\operatorname{HF}^{-}_{\rm
red}(\overline{\Sigma}_{\ell})$ is supported in odd degree (see Equation
(A.1)). By the Künneth formula [OS04a, Corollary 6.3] we can compute
$\operatorname{HF}^{-}_{\mathrm{red}}(Y_{\ell})$ (see Equation (A.2)), and by
taking the even-degree summand we obtain

(5.1)
$\operatorname{HF}^{-}_{\mathrm{red,even}}(Y_{\ell})\cong\left(\operatorname{HF}^{-}_{\mathrm{red}}(\Sigma_{\ell})\otimes\left(\operatorname{HF}^{-}_{\mathrm{red}}(\Sigma_{\ell})\right)^{*}\right)\oplus\operatorname{HF}^{-}_{\mathrm{red}}(\Sigma_{\ell}).$

(Note that there is a degree shift by $+1$ in
$\operatorname{HF}_{\textrm{red}}$ when changing the orientation, and a degree
shift by $-1$ when applying the Tor functor in the tensor product, and they
cancel out.)

Since

(5.2)
$\mathbb{F}[U]/(U^{k})\otimes_{\mathbb{F}[U]}\mathbb{F}[U]/(U^{j})\cong\mathbb{F}[U]/(U^{\min\\{k,j\\}}),$

the length of the longest tower in
$\operatorname{HF}^{-}_{\mathrm{red}}(\Sigma_{\ell})$ will be preserved in
$\operatorname{HF}^{-}_{\mathrm{red,even}}(Y_{\ell})$ too, hence showing
property (1).

We now turn to property (2). Referring to Equation (5.1), towers of length 1
with positive Maslov degree can only come from the first summand (i.e., the
tensor product), because $\operatorname{HF}^{-}_{\mathrm{red}}(\Sigma_{\ell})$
is supported in non-positive degree. On the other hand, the first summand is
symmetric, so it is enough to find the smallest Maslov grading in absolute
value that supports a tower of length $1$.

By Equation (5.2), towers of length $1$ in the first summand are obtained by
tensoring a tower of length $1$ with a tower of any length, and their Maslov
degree is the distance (in Maslov degree) between the generators of those two
towers. From the properties above, the minimum such distance in absolute value
is $(\ell-1)\ell-(\ell-2)(\ell-1)=2(\ell-1)$, thus proving property (2) for
$Y_{\ell}$.

Thus, $Y_{\ell}$ is of size $\ell$ for each $\ell\geq 2$. Since the Poincaré
homology sphere is an L-space, we also have that
$Y_{\ell}\\#n(P\\#\overline{P})$ has size $\ell$ for each $\ell\geq 2$.

We can now prove Proposition 5.1.

###### Proof of Proposition 5.1.

Choose $m,n$ distinct integers and suppose that the link $L_{K_{n}}$ in
$S^{4}$ is smoothly split by $Y_{K_{m}}$. Then, by Theorem 1.3,
$\operatorname{HF}_{\rm red}(Y_{K_{n}})$ is a direct summand of
$\operatorname{HF}_{\rm red}(Y_{K_{m}})$. However, $Y_{K_{n}}$ is of size $n$
and $Y_{K_{m}}$ is of size $m$, by Example 5.3, so as observed in Remark 5.2
neither of their $\operatorname{HF}_{\rm red}$ can be a direct summand of the
other. ∎

##  Appendix A Heegaard Floer homology

We recall here the basic facts we need about Heegaard Floer homology. This was
an invariant defined by Ozsváth and Szabó in [OS04b, OS04a] that associates to
a 3-manifold a collection of groups and to cobordisms between them
homomorphisms between the corresponding groups. Given the context of the
paper, we will restrict to 3-manifolds that are integer homology spheres and
to 4-manifold cobordisms that are rational homology cobordisms. This will
greatly simplify the discussion and allow us to avoid spinc structures and
rational gradings. We will work over the field $\mathbb{F}$ with two elements.

If $Y$ is an integer homology sphere, one of the groups associated to $Y$ is
called $\operatorname{HF}^{-}(Y)$: this is a finitely generated
$\mathbb{Z}$-graded $\mathbb{F}[U]$-module, where $U$ is a formal variable of
degree $-2$. The degree is called the _Maslov grading_. It sits into an exact
sequence of $U$-equivariant degree-$0$ maps222A word of caution here. There is
also an $\operatorname{HF}^{+}_{\mathrm{red}}$, which is a quotient of
$\operatorname{HF}^{+}$, and it differs from $\operatorname{HF}^{-}$ by a
grading shift: $\operatorname{HF}^{-}_{\rm red}\cong\operatorname{HF}^{+}_{\rm
red}[+1]$ as graded $\mathbb{F}[U]$-modules. When we use
$\operatorname{HF}_{\rm red}$ we refer to either of them, but we assume that
the choice we make is coherent.:

$0\to\operatorname{HF}^{-}_{\rm
red}(Y)\to\operatorname{HF}^{-}(Y)\to\mathcal{T}^{-}_{d(Y)}\to 0,$

where $\mathcal{T}^{-}$ is an _infinite tower_ ,
$\mathcal{T}^{-}\cong\mathbb{F}[U]$, $d(Y)$ is an even integer and
$1\in\mathcal{T}^{-}_{d(Y)}$ sits in degree $d(Y)$, and
$\operatorname{HF}^{-}_{\rm red}(Y)$ is a finitely-generated
$\mathbb{F}[U]$-module that is finite-dimensional over $\mathbb{F}$. In
particular, $\operatorname{HF}^{-}_{\rm red}(Y)$ splits as a direct sum of
_finite towers_ :

$\operatorname{HF}^{-}_{\rm
red}(Y)\cong\bigoplus_{i}(\mathbb{F}[U]/(U^{\ell_{i}}))_{d_{i}},$

for some (not necessarily even) integers $d_{i}$ and some positive integers
$\ell_{i}$. Here the $d_{i}$ signifies that the grading of
$1\in\mathbb{F}[U]/(U^{\ell_{i}})$ is $d_{i}$. (This is also the highest
degree among all elements.) The quantity $d(Y)$ is called the $d$-_invariant_
or _correction term_ of $Y$.

We denote with $\operatorname{HF}^{-}_{\mathrm{even}}(Y)$ (resp.
$\operatorname{HF}^{-}_{\mathrm{red,even}}(Y)$) the even-degree summand of
$\operatorname{HF}^{-}(Y)$ (resp. $\operatorname{HF}^{-}_{\mathrm{red}}(Y)$).
Since $U$ has degree $-2$, $\operatorname{HF}^{-}_{\mathrm{even}}(Y)$ is an
$\mathbb{F}[U]$-submodule of $\operatorname{HF}^{-}(Y)$.

There is a formula relating $\operatorname{HF}^{-}(Y)$ and
$\operatorname{HF}^{-}(\overline{Y})$: $d(\overline{Y})=-d(Y)$ and

(A.1)
$\operatorname{HF}^{-}_{\mathrm{red}}(\overline{Y})=(\operatorname{HF}^{-}_{\mathrm{red}}(Y))^{*}[+1],$

where the $*$ indicates that the degrees and the $U$-action are reversed, and
the $[+1]$ indicates that there’s a degree shift (this follows from [OS04a,
Proposition 2.5] and the fact that $\operatorname{HF}^{-}_{\rm
red}\cong\operatorname{HF}^{+}_{\rm red}[+1]$). So, for instance, to a finite
tower $(\mathbb{F}[U]/(U^{\ell_{i}}))_{d_{i}}$ in $\operatorname{HF}^{-}(Y)$
there corresponds a finite tower
$(\mathbb{F}[U]/(U^{\ell_{i}}))_{1-2\ell_{i}-d_{i}}$ in
$\operatorname{HF}^{-}(\overline{Y})$.

There is a Künneth formula relating
$\operatorname{HF}^{-}(Y),\operatorname{HF}^{-}(Y^{\prime})$, and
$\operatorname{HF}^{-}(Y\\#Y^{\prime})$: if we denote
$H=\operatorname{HF}^{-}_{\mathrm{red}}(Y)$ and
$H^{\prime}=\operatorname{HF}^{-}_{\mathrm{red}}(Y^{\prime})$, then by [OS04a,
Corollary 6.3]

(A.2) $\operatorname{HF}^{-}_{\mathrm{red}}(Y\\#Y^{\prime})\cong
H[d(Y^{\prime})]\oplus H^{\prime}[d(Y)]\oplus\left(H\otimes
H^{\prime}\right)\oplus\left(H\otimes H^{\prime}\right)[-1].$

If $W$ is a rational homology cobordism from $Y$ to $Y^{\prime}$, then there
is an induced homomorphism of $\mathbb{F}[U]$-modules

$F^{-}_{W}\colon\operatorname{HF}^{-}(Y)\to\operatorname{HF}^{-}(Y^{\prime})$

which has degree $0$, which descends to an analogous homomorphism in
$\operatorname{HF}^{-}_{\mathrm{red}}$. This association is functorial: if
$W^{\prime}$ is another rational homology cobordism from $Y^{\prime}$ to
$Y^{\prime\prime}$, then the composition of the associated maps is the map
associated to the cobordism $W^{\prime\prime}$ obtained by stacking $W$ and
$W^{\prime}$. In other words,

$F^{-}_{W^{\prime\prime}}=F^{-}_{W^{\prime}}\circ
F^{-}_{W}\colon\operatorname{HF}^{-}(Y)\to\operatorname{HF}^{-}(Y^{\prime\prime}).$

Finally, we have that $F^{-}_{I\times
Y}=\mathrm{id}_{\operatorname{HF}^{-}(Y)}$.

## References

  * [DLVW22] Aliakbar Daemi, Tye Lidman, David Shea Vela-Vick, and C-M Michael Wong.  Ribbon homology cobordisms.  Adv. Math., 408:108580, 2022.
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  * [LLP23] Adam Simon Levine, Tye Lidman, and Lisa Piccirillo.  New constructions and invariants of closed exotic 4-manifolds.  arXiv preprint arXiv:2307.08130, 2023.
  * [OS04a] Peter Ozsváth and Zoltán Szabó.  Holomorphic disks and three-manifold invariants: properties and applications.  Annals of Mathematics, pages 1159–1245, 2004.
  * [OS04b] Peter S. Ozsváth and Zoltán Szabó.  Holomorphic disks and topological invariants for closed three-manifolds.  Ann. of Math. (2), 159(3):1027–1158, 2004.
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# Towards Better Serialization of Tabular Data for Few-shot Classification
with Large Language Models

Sukriti Jaitly Tanay Shah Ashish Shugani Razik Singh Grewal
Carnegie Mellon University
<EMAIL_ADDRESS>
###### Abstract

We present a study on the integration of Large Language Models (LLMs) in
tabular data classification, emphasizing an efficient framework. Building upon
existing work done in TabLLM Hegselmann et al. (2023), we introduce three
novel serialization techniques, including the standout LaTeX serialization
method. This method significantly boosts the performance of LLMs in processing
domain-specific datasets, Our method stands out for its memory efficiency and
ability to fully utilize complex data structures. Through extensive
experimentation, including various serialization approaches like feature
combination and importance, we demonstrate our work’s superiority in accuracy
and efficiency over traditional models.

##  1 Introduction

Advancements in deep learning have significantly impacted fields like natural
language processing and computer vision, but tabular data hasn’t seen a
parallel rise Borisov et al. (2022). This disparity stems from tabular data’s
unique characteristics, such as mixed data types and fewer columns.

Large language models (LLMs) like GPT-3 Brown et al. (2020), with their
extensive pre-training, have shown prowess in tasks ranging from few-shot text
classification to tabular data cleaning. Recognizing this potential, TabLLM
was introduced as a bridge between LLMs and tabular data classification,
leveraging serialized natural-language representations of tabular rows for
classification. Initial evaluations highlighted TabLLM’s competitive edge over
other models, including traditional ones like gradient-boosted trees
Hegselmann et al. (2023).

Recognizing the potential of LLMs in tabular data classification, the TabLLM
framework was introduced. This framework leverages serialized natural-language
representations of tabular rows for classification, and has shown a
competitive edge over traditional models like gradient-boosted trees
Hegselmann et al. (2023). Building upon the success of TabLLM, our study
introduces a novel framework, which enhances the performance of LLMs in
tabular data classification through advanced serialization methods. Our
approach significantly diverges from previous models by utilizing LaTeX
serialization, which improves memory efficiency and the model’s ability to
process complex data structures.

Our project delves into various serialization approaches, including feature
combination and importance, to demonstrate the superiority of our techniques
in terms of accuracy and efficiency. We also explore the integration of key
tabular features into the LLM framework, enhancing its interpretability and
effectiveness in classification tasks. This study presents a comprehensive
examination of using novel serialisation techniques in tabular data
classification, highlighting its potential to transform data processing across
various industry areas.

##  2 Related Work

TabLLM: TabLLM is a novel framework for few-shot classification of tabular
data using large language models (LLMs). This approach focuses on converting
tabular data into natural language representations, then employing LLMs,
particularly the T0 model and GPT-3, for classification tasks. This method
addresses challenges in obtaining sufficient labeled data for traditional
supervised learning, especially in fields like healthcare.

It explores different methods for serializing tabular data into natural
language inputs for language models, focusing on improving zero and few-shot
classification performance. Nine serialization formats are studied, ranging
from simple list templates to more complex methods like Table-To-Text and
GPT-3 based serializations. The paper also examines ablations like listing
only values or permuting names/values to understand the impact of column names
and the precision of feature values on classification performance.

TabLLM outperforms traditional deep-learning methods and rivals gradient-
boosted trees, particularly in few-shot learning scenarios. TabLLM’s efficacy
is highlighted by its ability to leverage the extensive knowledge encoded in
pre-trained LLMs, requiring minimal labeled data. The framework’s adaptability
to different types of tabular data and its competitive performance against
traditional models underline its potential in domains where labeled data is
scarce or expensive to obtain.

TabLLM requires significant computational resources, including large GPUs for
fine-tuning, and its performance may decline if a dense feature set exceeds an
LLM’s token limit. Despite these drawbacks, TabLLM shows excellent performance
in tabular classification, especially in few-shot scenarios, outperforming
algorithms like XGBoost and SAINT.

TabNet: TabNet is a novel high-performance and interpretable canonical deep
tabular data learning architecture which uses sequential attention to choose
which features to reason from at each decision step, enabling interpretability
and more efficient learning as the learning capacity is used for the most
salient features. It stands out from its competitors in the following ways:
1\. It is built to work on raw tabular data without the necessity for
preprocessing, allowing for gradient descent-based optimization and seamless
integration with end-to-end learning.
2\. A distinctive feature of TabNet is its sequential attention mechanism,
which determines the features to focus on during each decision step.
3\. TabNet’s design choices endow it with commendable advantages. Empirical
studies reveal that it either surpasses or matches the performance of other
tabular learning models across diverse datasets and problem types.
4\. Another pioneering feature of TabNet is its capability to harness the
power of unsupervised pre-training for tabular data, which leads to remarkable
performance enhancements by predicting masked features.

TabLLM and TabNet are both related works in the field of tabular data analysis
and machine learning, but they approach the problem from different angles and
have distinct characteristics. While both TabLLM and TabNet deal with tabular
data, they differ in their core objectives and methodologies. TabLLM
emphasizes few-shot classification using large language models, making it
suitable for scenarios with limited labeled data. In contrast, TabNet
prioritizes interpretability and efficient learning, focusing on creating a
versatile and competitive deep learning model for tabular data. Their
relatedness lies in their common goal of advancing the capabilities of machine
learning in the domain of tabular data analysis, but they take distinct paths
to achieve it.

##  3 Method

###  3.1 Serialization

Leveraging Large Language Models (LLMs) for tabular data necessitates
converting tables into a format akin to natural language. This procedure
involves serializing table content into coherent text while incorporating
task-specific cues into the prompt. Our approach tested various serialization
techniques, emphasizing the significance of features, their combinations, and
structuring the input with a LaTeX framework to align with LLMs’ textual
processing capabilities.

###  3.2 Fine-tuning

We implemented fine-tuning on the T0 (3 billion) model Sanh et al. (2021)
using the Intrinsic Attention-based Prompt Tuning (IA3) Liu et al. (2022) as a
Parameter-Efficient Fine-Tuning (PEFT) technique. The approach was applied to
the serialized dataset across a broad spectrum of training examples, ranging
from zero-shot to 32 shots. This detailed exploration covered various levels
of data scarcity, allowing us to rigorously evaluate the model’s adaptability
and performance across a gradient of data availability scenarios.

###  3.3 LLM for Prediction

Given an LLM with a vocabulary $V$, the output prompted by the LLM, denoted as
$\text{LLM}(\text{serialize}(F,x),p)\in V^{*}$, is derived from the natural-
language input serialization of the features $F$ and instance $x$ combined
with a prompt $p$. In a few-shot learning context, the set
$\\{(\text{serialize}(F,x),p)\mid(x,y)\in D_{k}\\}$ is utilized for fine-
tuning the LLM. The generated text in the vocabulary space $V^{*}$ must then
be translated into a valid class label in $C$. Existing methods, such as the
verbalizer approach, establish this mapping from LLM output tokens to discrete
label spaces. We adopt a manual specification for this mapping in our work.

Figure 1: Overview of Framework

##  4 Experimental Setup

###  4.1 Dataset Details

In our research, we utilized the Vehicle Claims dataset, introduced by Chawda
et al. (2022) in "Unsupervised Anomaly Detection for Auditing Data and Impact
of Categorical Encodings". This dataset focuses on fraudulent insurance claims
in the automotive sector, characterized by a high number of categorical
attributes. It includes detailed information on vehicle make, model, color,
registration year, body type, engine size, gearbox, fuel type, and various
numerical attributes like mileage, price, seat and door numbers, along with
the type and specifics of vehicle issues, repair complexity, duration, and
cost. The dataset is available on GitHub.

###  4.2 Baseline

TabLLM investigated multiple approaches to serialize tabular data for language
model processing, with the Text-template method emerging as the most
effective. Therefore, TabLLM with Text-template serialization was set as a
baseline. The Text Template serialization in TabLLM involves converting
features and values into a natural language string where each column name is
followed by its corresponding value. This format mimics how humans might
naturally describe the data.

For example, a serialized entry could read, "Age is 35, sex is female, race is
Caucasian," and so on, adding relevant visit information like date, doctor
type, and primary complaint. This method respects the token limit of the LLM,
ensuring that a manageable amount of information is processed.

In the test AUC performance results, TabLLM generally outperforms or is
competitive with XGBoost and TabPFN across various datasets (serialized using
Text-template) and numbers of shots, especially in the very-few-shot regime.

###  4.3 Serialization with Feature Combination

In refining TabLLM, we introduced a feature combination technique, moving away
from the traditional single-feature-per-sentence format. This new approach is
designed to align closely with natural language, enhancing the model’s ability
to interpret and comprehend the complex interrelationships between different
features in tabular data. By transitioning from a simplistic serialization
template, which treated each feature in an isolated way, to a more integrated
format, we mimic the natural flow of human language.
For example, in our vehicle claims dataset, attributes like make, color and
body type are now combined into a single richer sentence.

###  4.4 Serialization with Feature Importance

A critical part of our approach involves understanding the significance of
different features in our dataset. To achieve this, we employed a covariance-
based method to identify the most influential features with respect to the
target variable, ’Label’. The process involved the following steps:

  1. 1.

Conversion of categorical data into numerical format using one-hot encoding.

  2. 2.

Computation of covariance values between all features and the target variable.

  3. 3.

Ranking of features based on the absolute values of their covariance and
selection of the top 4.

  4. 4.

Two techniques were then explored for enhancing feature importance
recognition:

    1. (a)

Adding the prefix “Critically” to the most important features of the prompt.

    2. (b)

Adding a sentence at the end of the original prompt as: “The 4 most important
features for the inference are: A, B, C, D.”

This method provided us with a clearer picture of which features our model
should prioritize during classification, in an attempt to enhance the accuracy
and efficiency of the LLM in handling tabular data.

###  4.5 LaTeX Serialization

Moving away from the traditional table-to-text serialization techniques, we
explored the novel approach of converting tabular data into LaTeX code format.
This LaTeX representation of the table was then used as the input for fine-
tuning our LLM by just passing a row representation preceded by hline tag
without any headers. This method is grounded in the hypothesis that such a
structured representation, familiar in academic and scientific publications,
might aid the LLM in capturing the nuances of tabular data more effectively.
The LaTeX format, being more structured and consistent than plain text, could
potentially allow the LLM to understand and process the tabular data with
higher accuracy, especially in complex dataset where traditional serialization
methods may fall short.

##  5 Results

TabLLM | Number of Shots
---|---
Serialization Method | 0 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512
XGBoost | _ | 0.5 | 0.5 | 0.441 | 0.617 | 0.579 | 0.800 | 0.912 | 0.946
Text Template (Baseline) | 0.485 | 0.495 | 0.512 | 0.557 | 0.642 | 0.954 | 0.985 | 0.994 | 0.997
Feature Importance | 0.477 | 0.491 | 0.499 | 0.54 | 0.653 | 0.942 | 0.944 | 0.972 | 0.982
Feature Combination | 0.509 | 0.521 | 0.531 | 0.582 | 0.665 | 0.933 | 0.986 | 0.992 | 0.997
LaTeX | 0.686 | 0.684 | 0.652 | 0.666 | 0.778 | 0.962 | 0.986 | 0.990 | 0.992
Table 1: AUC for varying number of training examples (shots)

Figure 2 shows the performance of various serialization techniques compared
with the traditional Machine Learning model, XGBoost Chen and Guestrin (2016),
and the best performing serialization technique presented in TabLLM. In zero-
shot setting LaTeX serialization performs significantly well when compared to
other three serialization techniques that performs similar to the XGBoost.

Although, the LaTeX serialization shows small dip in performance with
additional in-context examples, it still outperforms other methods. Our
hypothesis for the dip in performance is that the selection of examples is not
the best set of examples for the model to learn. As we increase the number of
shots, we see an increase in the performance with all the techniques and after
64-shots, we get similar performance. These results are also show in Table 1.

Notably, we were constrained to a maximum of 24 GB of GPU when utilizing the
highest available AWS instance (g5.2xlarge). Due to this constraint and the
memory usage of T03B model, when using all 17 columns with serialization with
manual template, feature importance and feature combination we observer that
after 32-shots we face CUDA Out of Memory error. To mitigate this problem, we
eliminated few trivial columns based on feature importance from the dataset
and experimented with the same. However, while working with LaTeX
serialization, we used a single row represented by hline tag and the template
did not contain any headers related to the table due to which we did not face
the memory issue and hence we were able to fit all the columns. Hence, LaTeX
serialization not only gives better performance but also provides memory
efficiency.

Figure 2: AUC for the anomaly dataset with different number of training
examples (shots)

##  6 Discussion

One of the most interesting aspects of our project is our approach to model
tuning. We employed the parameter-efficient fine-tuning technique. Initially,
we hadn’t anticipated the need for such a technique, but it became evident
that to draw parallels with TabLLM behavior and draw results with the given
computational budget, we have to employ those techniques.
A significant challenge we encountered was the computational restrictions,
which posed limitations on the scale of our dataset. Tuning the model with our
chosen dataset proved to be more demanding than we had anticipated. As a
result, the experiments we have conducted so far are based on a relatively
small subset of data. This limitation is crucial to note, as the results
derived from such a limited dataset might not be comprehensive and that means
that further research and evaluations on more extensive datasets are necessary
to validate and generalize our results.

##  7 Conclusion

This study has successfully demonstrated the innovative application of Large
Language Models (LLMs) in the domain of tabular data classification, with a
specific focus on our new LaTeX serialisation framework. Our approach, which
builds upon the foundational TabLLM Hegselmann et al. (2023) techniques, has
introduced novel serialization methods that have proved effective in handling
domain-specific datasets. Among these, the LaTeX serialization method stands
out, leveraging a structured LaTeX code format for tabular data
representation. This method not only enhances LLM performance in
classification tasks but also exhibits remarkable memory efficiency and
computational efficacy, enabling the complete utilization of all data columns
without encountering computational constraints like CUDA Out of Memory errors.
Additionally, our exploration of other serialization techniques, such as
integrating multiple features into coherent sentences and emphasizing feature
importance, has furthered our understanding of LLMs’ capabilities in
interpreting complex data relationships. While these methods played a
secondary role to the primary focus on LaTeX serialization, they collectively
contribute to a more nuanced understanding of how LLMs can be effectively
applied to diverse data scenarios. This study, therefore, not only underscores
the potential of the LaTeX serialization in the realm of LLM-driven tabular
data analysis but also sets the stage for future research in developing more
sophisticated and efficient tools for data processing across various industry
domains.

##  8 Societal Implications

The application of LLMs in tabular data classification has far-reaching
implications in various societal domains. By enhancing the accuracy and
efficiency of such models, we can significantly improve decision-making
processes in areas like healthcare, finance, and environmental studies. The
ability of LLMs to adapt to different domains and handle sparse data can lead
to more informed and data-driven solutions to critical societal issues.

## References

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# Geometric representation of adiabatic distributed-Bragg-reflectors and
broadening the photonic bandgap

Shailja Sharma
School of Physical Sciences
National Institute of Science Education and Research
HBNI, Jatni - 752050, Odisha, India
<EMAIL_ADDRESS>
&Abhishek Mondal
School of Physical Sciences
National Institute of Science Education and Research
HBNI, Jatni - 752050, Odisha, India
<EMAIL_ADDRESS>
&Ritwick Das*
School of Physical Sciences
National Institute of Science Education and Research
HBNI, Jatni - 752050, Odisha, India
<EMAIL_ADDRESS>

###### Abstract

Adiabatic following has been an widely-employed technique for achieving near-
complete population transfer in a ‘two-level’ quantum mechanical system. The
theoretical basis, however, could be generalized to a broad class of systems
exhibiting SU(2) symmetry. In the present work, we present an analogy of
population transfer dynamics of two level atomic system with that of light
propagation in a classical ‘one-dimensional’ photonic crystal, commonly known
as distributed-Bragg-reflector (DBR). This formalism facilitates in adapting
the idea of adiabatic following, more precisely the rapid adiabatic passage
(RAP) which is usually encountered in a broad class of quantum-mechanical
systems. We present a chirped DBR configuration in which the adiabatic
constraints are satisfied by virtue of optimally chirping the DBR. The
reflection spectrum of the configuration exhibit broadening of photonic
bandgap (PBG) in addition to a varying degree of suppression of sharp
reflection peaks in the transmission band. The intermodal coupling between
counter-propagating modes as well as their phase-mismatch, for the DBR
configuration, exhibits a longitudinal variation which is usually observed in
‘Allen-Eberly’ scheme of adiabatic population transfer in _two-level_ atomic
systems.

##  1 Introduction

The propagation characteristics of electromagnetic ($em$) waves in a
periodically stratified dielectric medium closely resemble the features
exhibited by the matter waves in crystalline solids [1, 2]. In crystals, the
periodic _Coulomb_ -potential leads to formation of continuous energy bands
separated by forbidden energy spectrum, also known as bandgaps. In
resemblance, analogous periodic photonic architectures, commonly termed as
_photonic crystals_ (PCs), lead to formation of $em$ transmission _bands_
enclosed by forbidden frequency spectrum known as photonic bandgap (PBG) [3,
4]. Over the last four decades, PCs has evolved as backbone of many
technological advancements which essentially hinge upon manipulating the
spatial and spectral characteristics of light beam [5]. In one-dimension, the
PCs are also known as distributed-Bragg-reflector (DBR) and they have turned
out to be primary ingredients for devising reflectors/anti-reflectors,
spectral and spatial filtering elements and creating efficient sensing
platforms which include those involving plasmonic interactions [6, 7, 8, 9,
10, 11]. Bragg reflection based waveguiding configurations have been employed
as polarization selection device in miniaturized optical sources [12, 13].
Bragg-reflection-waveguides (BRWs) with a vacuum core have been forecasted as
plausible hosts for future optical particle accelerators driven by extremely
high power lasers [14]. On the other hand, the possibility of field
confinement and therefore, reducing the $em$ interaction volume to sub-
wavelength scale has lead to the formation of stable spatial-solitons in
nonlinear BRWs [15]. Recently, the bulk modes of a BRWs or optical surface
states in DBR have been found to be promising candidates for optical
parametric processes and higher harmonic generation [16, 17, 18]. As mentioned
before, a DBR is primarily characterized by PBG whose magnitude is primarily
dictated by the refractive index contrast of the DBR constituents and the
location in a spectral band is governed by the thickness of constituent layers
[5]. Therefore, for a given pair of material forming the DBR, the magnitude of
PBG is unique and fixed. Within the limit of optical transparency for the
constituent DBR materials, we present a formalism to broaden the PBG (and
suppress the transmission band) by utilizing the concept of adiabatic coupling
to a backward propagating mode from a forward propagating mode. Therefore, we
draw an equivalence of the coupled-wave equations in a DBR with that
encountered while describing the dynamics of quantum _two-level_ atomic
systems. Subsequently, we propose plausible DBR configurations for adopting
the adiabatic following in such systems which leads to broadening of PBG
spectrum. The formalism could be translated to any spectral band and PBG
broadening is limited by the restriction imposed by material transparency
only. Interestingly, this idea provides an viable platform to tailor the
backscattered phase from such systems as well.

Adiabatic following, also known as rapid adiabatic passage (RAP) has been an
well-established technique for realizing near $100\%$ population transfer in a
_two-level_ atomic systems using optimum ultrashort ($\leq~{}0.5~{}ps$) pulses
[19]. In fact, it is quite straightforward to ascertain that the RAP mechanism
could be adopted in any equivalent system exhibiting _special unitary(2)_ or
_SU(2)_ symmetry. For example, RAP provides an efficient platform for
ultrashort pulse frequency conversion as well as tailoring spatial
characteristics of optical beams in optical nonlinear medium [20, 21, 22, 23].

##  2 Adiabatic-following in DBR

A periodic variation (with periodicity $\Lambda$) in dielectric constant
($\epsilon$) along $z$-direction in a medium could be expressed as [24]

$\epsilon(x,y,z)=\epsilon(x,y)+\Delta\epsilon(x,y,z)$ (1)

where $\Delta\epsilon(x,y,z)$ defines the periodic modulation along $z$. This
dielectric perturbation gives rise to the possibility of intermodal
interactions. Let us consider any two modes, namely $\ket{p}$ and $\ket{q}$
propagating through the medium described by dielectric function in equation
(1). Assuming the modes are propagating along $z$-direction, the $em$-field
corresponding to the modes (in the paraxial limit) are given by
$\ket{p}~{}=~{}E_{p}(x,y)A_{p}(z)e^{i(\omega t-\beta_{p}z)}$ and
$\ket{q}~{}=~{}E_{q}(x,y)A_{q}(z)e^{i(\omega t-\beta_{q}z)}$. Under the slowly
varying approximation, the evolution of $z$-dependent mode-amplitude ($A_{p}$)
of the $p^{th}$-mode due to the presence of $q^{th}$-mode (with amplitude
$A_{q}$) is given by [24]

$\frac{dA_{q}}{dz}=-i\frac{\beta_{q}}{|\beta_{q}|}\sum_{p}\sum_{m}{\kappa_{pq}}^{(m)}A_{p}e^{-i(\beta_{p}-\beta_{q}-m\frac{2\pi}{\Lambda})z}$
(2)

where $\beta_{p}$ and $\beta_{q}$ are the normal components of wavevector
$k_{p}$ and $k_{q}$ respectively. ${\kappa_{pq}}^{(m)}$ defines the magnitude
of coupling coefficient which couples the modes through the $m^{th}$ Fourier
component of dielectric function equation (1). The $m^{th}$ coupling
coefficient is expressed as

${\kappa_{pq}}^{(m)}=\frac{\omega}{4}\int\int{E^{*}}_{p}(x,y)\epsilon_{m}(x,y){E}_{q}(x,y)dxdy$
(3)

where $\epsilon_{m}$ is the $m^{th}$ component of Fourier-series expansion of
$\epsilon$ (see equation (1)). The coupling between the interacting modes is
maximum when the longitudinal phase-matching condition is exactly satisfied,

$\beta_{p}-\beta_{q}-m\frac{2\pi}{\Lambda}=0$ (4)

In case of DBR, the coupling between a forward propagating mode ($A_{p}\equiv
A_{i}$) and a backward propagating mode ($A_{q}\equiv A_{r}$) is of interest.
In absence of any other intermodal interactions, the forward-backward mode
coupling (assuming $m=1$) would be governed by [24, 25, 26],

$\frac{dA_{r}}{dz}=-i\frac{\beta_{r}}{|\beta_{r}|}{\kappa}A_{i}e^{i\Delta\beta
z}$ (5)
$\frac{dA_{i}}{dz}=-i\frac{\beta_{i}}{|\beta_{i}|}{\kappa^{*}}A_{r}e^{-i\Delta\beta
z}$ (6)

where we have dropped the indices from $\kappa$ and
$\Delta\beta=\beta_{r}-\beta_{i}-\frac{2\pi}{\Lambda}$. In case the DBR
constituent materials are isotropic, the dielectric function (equation (1)) is
a purely scalar and consequently, transverse-electric ($TE$) to transverse-
magnetic ($TM$) mode-coupling (and vice-versa) is forbidden. It is worth
noting that the phase-matching condition for coupling the forward propagating
mode ($\ket{i}$) to a backward propagating mode ($\ket{r}$) or vice-versa is
contra-directional in nature and therefore, Eq. (4) could be expressed as
$\Delta\beta=2\beta-\frac{2\pi}{\Lambda}=0$ where $\beta_{i}=-\beta_{r}=\beta$
[24]. As shown in Fig.1, the phase-matching condition for the contra-
directional coupling process is modified to
$\Delta\beta=2\beta\cos{\theta}-\frac{2\pi}{\Lambda}=0$ in case of oblique
incidence ($\theta$ with respect to $z$-axis),. For a simple case when the two
layers of the DBR (with refractive indices $n_{1}$ and $n_{2}$) share the same
thickness _i.e._ $d_{1}~{}=~{}d_{2}$, the coupling coefficient takes a
simplified form given by [24],

$\kappa_{TE}=\frac{i}{\lambda\cos{\theta}}\frac{\sqrt{2}({n_{1}}^{2}-{n_{2}}^{2})}{\sqrt{({n_{2}}^{2}+{n_{1}}^{2})}}$
(7)
$\kappa_{TM}=\frac{i}{\lambda\cos{\theta}}\frac{\sqrt{2}({n_{1}}^{2}-{n_{2}}^{2})}{\sqrt{({n_{2}}^{2}+{n_{1}}^{2})}}\cos{2\theta}$
(8)

The solution of equations (5) and (6) for _close-to-perfect_ phase-matched
($\Delta\beta~{}\approx~{}0$) situation leads to a strong coupling from a
forward propagating mode ($\ket{i}$) to a backward propagating mode
($\ket{r}$) for broad range of frequencies. This is realized for all the
frequencies within the PBG and consequently, we obtain a strong reflection
band. It is apparent from equations (7) (or (8)) that the contrast in
refractive index (or dielectric constant) is the primary factor determining
the strength of coupling and hence, the sharpness of band edges. For a given
choice of DBR constituents, $|n^{2}_{1}-n^{2}_{2}|$ is fixed and consequently,
the width of PBG remains unchanged.

Figure 1: Oblique incidence of $em$-wave on distributed-Bragg-reflector (DBR).
$z$-direction represents the optical axis

###  2.1 Adiabatic phase-matching

We consider the following transformation to a rotating frame given by

$A_{r}=\tilde{a_{r}}e^{i/2[\Delta\beta(0)z-{\int_{0}}^{z}q(\tilde{z})d\tilde{z}]}$
(9)
$A_{i}=\tilde{a_{i}}e^{-i/2[\Delta\beta(0)z-{\int_{0}}^{z}q(\tilde{z})d\tilde{z}]}$
(10)

where $\Delta\beta(z)=\Delta\beta(0)z-{\int_{0}}^{z}q(\tilde{z})d\tilde{z}$
and $\Delta\beta(0)$ is the phase-mismatch at $z=0$, $q(z)$ is determined by
the dielectric constant modulation $\Delta\epsilon$ (as per Eq. (1)), and
$\Delta k=\frac{q(z)-\Delta\beta(0)}{2}$. The complex amplitudes
$\tilde{a_{i}}$ and $\tilde{a_{r}}$ represent the power contained in the
forward and backward propagating modes in the rotated reference frame. It is
worthwhile to note that $|\tilde{a_{i}}|^{2}=|{A_{i}}|^{2}$ and
$|\tilde{a_{r}}|^{2}=|{A_{r}}|^{2}$. The substitution of equation (9) and (10)
into equations (5) and (6) yields

$-i\frac{d}{dz}\begin{pmatrix}\tilde{a_{r}}\\\ \tilde{a_{i}}\\\
\end{pmatrix}=\begin{pmatrix}\Delta k&\kappa\\\ -\kappa^{*}&-\Delta k\\\
\end{pmatrix}\begin{pmatrix}\tilde{a_{r}}\\\ \tilde{a_{i}}\\\ \end{pmatrix}$
(11)

which is an equivalent representation of time-dependent Schrödinger equation
in optics. In fact, if we define a state
$\big{|}\Psi\big{>}=\bigg{(}\begin{matrix}\tilde{a_{r}}\\\
\tilde{a_{i}}\end{matrix}\bigg{)}$ then
$-i\frac{d}{dz}\ket{\Psi}=\hat{H}\ket{\Psi}$ where time-dependence is replaced
by the $z$-dependence and the Hamiltonian $\hat{H}=\vec{\sigma}.\vec{B}$.
Here, $\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli-spin
matrices and $\vec{B}=(\kappa,0,\Delta k)$ represents a fictitious magnetic
field. The Hamiltonian has a close resemblance with that representing the
dynamics of a spin-$1/2$ particle in an external magnetic field where
$\ket{i}$, $\ket{r}$ are the equivalent to the spin-up ($\ket{\uparrow}$) and
spin-down ($\ket{\downarrow}$) states[27]. In order to represent a particular
pseudo-spin state, the equivalent Stokes parameters could be given by
$S_{j}=\big{<}\sigma_{j}\big{>}$ ($j\equiv x,y,z$) and the evolution of the
state would be dictated by $\frac{\partial{S}_{j}}{\partial
z}=-i\big{<}\big{[}\sigma_{j},\hat{H}\big{]}\big{>}$ . This, subsequently,
leads to

$\begin{matrix}S_{x}=\tilde{a_{i}}{\tilde{a_{r}}}^{*}+\tilde{a_{r}}{\tilde{a_{i}}}^{*}\\\
S_{y}=-i[\tilde{a_{i}}{\tilde{a_{r}}}^{*}-\tilde{a_{r}}{\tilde{a_{i}}}^{*}]\\\
S_{z}={|\tilde{a_{r}}|}^{2}-{|\tilde{a_{i}}|}^{2}\end{matrix}$ (12)

and

$\dot{\vec{S}}=\vec{B}\times\vec{S}$ (13)

where $\dot{\vec{S}}\equiv\frac{\partial\vec{S}}{\partial z}$). The states and
their evolution described by equations (12) and (13) determine the evolution
dynamics of the forward and backward propagating modes. In other words,
equations (12) and (13) could be mapped onto an equivalent Bloch-sphere where
$\ket{i}$ (spin-up) and $\ket{r}$ (spin-down) states are located the _south_
-pole and _north_ -pole respectively. Any point of the DBR Bloch-sphere
represents a particular superposition of eigenstates $\ket{i}$ and $\ket{r}$.
_Sec. 2.2_ discusses the geometric representation in detail for specific cases
of evolution of these states. The fictitious magnetic-field ($\vec{B}$)
constitutes a parameter space for exerting a _torque_ on the system which
leads to precession of the state vector ($\vec{S}$) about $\vec{B}$ with a
frequency $|\vec{B}|=\sqrt{{|\kappa|}^{2}+{\Delta k}^{2}}$. It is worthwhile
to point out that the value $S_{z}$ (equation (12)) determines the mode-
conversion (from the forward-to-backward or vice-versa) factor. The state
$\vec{S}=[0,0,-1]$ represents a situation where the incident beam contains all
the optical power $i.e.~{}\tilde{a}_{i}=1$ and the state $\vec{S}=[0,0,1]$
represents the case when all the optical power is present in the reflected
beam ($\tilde{a}_{r}=1$). The conversion efficiency to the reflected beam is
expressed as $\eta=\frac{S_{z}+1}{2}$ as shown in Fig. 5. The dynamics, here,
is primarily controlled by the parameters $\Delta k$ and $\kappa$ which are
essentially equivalent to factors _detuning_ ($\Delta$) and _Rabi-frequency_
($\Omega$) respectively which are commonly encountered in a _two-level_ atomic
system. In a perfectly phase-matched case ($\Delta k=0$), which is satisfied
for the central PBG frequency, a gradual semi-circular rotation of the state-
vector ($\vec{S}$) from south-pole to the north-pole takes place on DBR Bloch-
sphere which implies a complete transfer of optical power from the incident
beam to the reflected beam in the DBR. In order not to obscure the salient
features of the equivalent dynamics, we have ignored the role played by
absorption loss ($\alpha$) in the dielectric materials. However, it would be
worthwhile to point out that the role of absorption loss is equivalent to that
played by semi-phenomenological decay constant ($T_{2}$) in a _two-level_
atomic system [19]. For $\Delta k\neq 0$, $\vec{S}$ traces a trajectory on the
DBR Bloch-sphere about the axis $\hat{n}=\frac{\kappa\hat{x}+\Delta
k\hat{z}}{\sqrt{\kappa^{2}+\Delta k^{2}}}$. This implies that $\Delta k\neq 0$
leads to transportation of state-vector $\vec{S}$ to a point in northern-
hemisphere. This point may be near the north-pole but it will never be the
north-pole of the equivalent DBR Bloch-sphere. Nevertheless, $S_{z}$ still
positive and the reflectivity is large (not maximum). The frequencies for
which $S_{z}$ remains positive forms a band of frequencies constituting the
PBG. However, a sufficiently large $\Delta k$ results in a $\hat{n}$ such that
$\vec{S}$ is transported to a point which lies on the southern hemisphere of
DBR Bloch-sphere (assuming $\vec{S}$ starts precessing about $\vec{B}$ from
south-pole). Therefore, $S_{z}$ would be negative after propagation through
the DBR. This corresponds to frequencies outside the PBG (or within the
transmission band) where reflectivity is much smaller than the PBG. In fact,
the _equator_ of DBR Bloch-sphere distinguishes the PBG (northern hemisphere)
from the transmission (pass) band (southern hemisphere) with respect to the
trajectory of $\vec{S}$.

It is apparent that $\Delta k=0$ along the entire DBR length for the central
PBG frequency and $\Delta k\neq 0$ for all other frequencies. The
aforementioned analogy to a _two-level_ atomic system, thus, allows us to
adopt the Rapid Adiabatic Passage (RAP) mechanism for broadening the PBG
spectrum and suppressing the transmission (pass) bands in the DBR.
Consequently, we propose DBR configurations which leads to _slow_ variation in
$\Delta k$ from a large negative to a large positive value along the DBR
length such that the sweep always remains much smaller than the coupling
($\kappa$). This would ensure nearly complete transfer of optical power from a
forward propagating to a backward propagating mode across a frequency band
extending much beyond the conventional PBG of a DBR. The RAP condition is
expressed as [19, 28]

$\kappa\dot{\Delta k}-\Delta k\dot{\kappa}<<{({|\kappa|}^{2}+{\Delta
k}^{2})}^{3/2}$ (14)

which essentially implies that the _rate_ of change of polar angle
$\Phi={tan}^{-1}(\frac{\kappa}{\Delta k})$ (angle between $\vec{B}$ and $z$
axis) during the evolution is much slower with the frequency of rotation
($=\sqrt{\kappa^{2}+\Delta k^{2}}$) for $\vec{S}$ about $\vec{B}$. If $\kappa$
is assumed to be constant (along $z$), the adiabaticity condition appears as
$\frac{d\Delta k}{dz}<<\frac{{(\kappa^{2}+{\Delta k}^{2})}^{3/2}}{\kappa}$.
Also, in order to achieve near-complete optical power transfer, it is
essential that the two states (modes) $\ket{i}$ and $\ket{r}$ are decoupled at
the entry ($z=0$) and exit ($z=L$) faces of the DBR. Alternately, this is
mathematically expressed as $|\frac{\Delta\beta}{\kappa}|~{}>>~{}2$ at $z=0,L$
which is equivalent to satisfying the condition of _autoresonance_ in ‘two-
wave’ interaction system [29, 30]. In this case, _autoresonance_ essentially
ensures that the counter-propagating modes remain phase-locked when the
parameters of the Hamiltonian undergo an adiabatic change. This manifests into
a near-complete transfer of optical power from the forward to the backward
propagating mode.

###  2.2 Chirped DBR configuration for adiabatic mode-conversion

Figure 2: A schematic to describe the geometry of a chirped-DBR

In order to adopt an adiabatic coupling scheme, we consider a DBR
configuration with a linear chirp in the duty cycle of each unit cell _i.e._
thickness $d_{1M}=d_{1}+M\delta$ and $d_{2M}=\Lambda-d_{1}-M\delta$ define the
thickness of layers $A$ and $B$ respectively in $M^{th}$ unit cell as shown in
Fig. 2. The unit cell period ($\Lambda$), however remains unchanged. Here,
$M=0,1,2,3,...,(N-1)$ where $N$ is the total number of unit cells in the DBR.
This leads to a longitudinal variation in $\Delta k$ through a monotonic
change in average refractive index of an _unit cell_ which could be defined as
$\bar{n}=\sqrt{\frac{d_{1M}{n_{1}}^{2}+d_{2M}{n_{2}}^{2}}{\Lambda}}$. It is
worthwhile to note that the variation of $\bar{n}$ also manifests in the form
$z$-dependence of $\kappa$. Using equations (4), (7) and (8), the variation in
$\Delta k$ and $\kappa$ for normal incidence ($\theta=0$) would be expressed
as [24]

$\Delta
k=\frac{\Delta\beta}{2}=\frac{2\pi\bar{n}}{\lambda}-\frac{\pi}{\Lambda}$ (15)
$\kappa=\frac{i(1-\cos{(\frac{2\pi
d_{1M}}{\Lambda}}))}{2\lambda}\frac{({n_{1}}^{2}-{n_{2}}^{2})}{\bar{n}}$ (16)

Figure 3: a) shows the variation of $\Delta\beta$ and $\kappa$ in $M^{th}$
unit cell. b) shows the variation of LHS and RHS of the inequality given in
Eq. (14) in $M^{th}$ unit cell. c) shows the variation of
$\frac{\Delta\beta}{\kappa}$ as a function of unit cell no. ($M$) for
depicting a significant fraction of DBR length satisfies the _(condition for
auto-resonance)_. d), e) and f) shows a comparison between the reflection
spectrum of a normal-DBR (dashed green line) of $d_{1}=d_{2}=200~{}nm$ with
that for C-DBRs (solid red line) having $\delta=~{}10~{}nm$,
$\delta=~{}5~{}nm$ & $\delta=~{}2.5~{}nm$ respectively.

For an arbitrarily chosen chirp-length of $\delta=10~{}nm$, $d_{1}=10~{}nm$
and $N=39$, the variation in $\Delta k$ and $\kappa$ for the chirped-DBR
(C-DBR) is shown in Fig. 3(a). It is apparent that $\Delta\beta$ ($=2\Delta
k$) varies symmetrically from a large negative (at $z=0$ or $M=0$) to a large
positive value (at $z=L$ or $M=38$). The coupling coefficient ($\kappa$), on
the other hand, reaches a maximum at the center of C-DBR geometry
($M~{}=~{}20$) and negligibly small at $z=0,L$. Figure 3(b) shows the
variation of $\kappa\frac{d\Delta\beta}{dz}-\Delta\beta\frac{d\kappa}{dz}$ in
$M^{th}$ unit cell of C-DBR. It is apparent that this is much smaller than
$(\kappa^{2}+\Delta\beta^{2})^{3/2}$ at any point within the C-DBR. Therefore,
the adiabaticity condition described by equation (14) is completely satisfied
in case of C-DBR. It is interesting to note that the auto-resonant condition
$i.e.~{}\big{|}\frac{\Delta\beta}{\kappa}\big{|}<2$ is satisfied in a
significant fraction of the C-DBR ($M\approx 10$ to $M\approx 28$) as shown in
Fig. 3(c). This essentially implies that the counter-propagating modes (at a
certain frequency) would be phase-locked in the entire interaction region if
they are phase-matched ($\Delta\beta=0$) in any one _unit cell_ of C-DBR.

The impact of closely satisfying the adiabatic constraints have a profound
impact on reflection spectrum shown in Fig. 3(d). In order to obtain the
reflection spectrum, we consider a cross-section as shown in Fig. 2 where
$N=39$ is considered. A broadband plane-wave is incident on the C-DBR from
$z=0$. The reflection spectrum was obtained using finite element method (FEM)
(wave-optics module, COMSOL Multiphysics). The periodic boundary condition is
imposed along the transverse direction and a mesh-size of $5~{}nm$ is
considered for the simulation. Also, we ignore the material dispersion in the
present case and assumed $n_{1}=2.5$ ($A\equiv TiO_{2}$) and $n_{2}=1.5$
($B\equiv SiO_{2}$) across the entire spectrum. In order to compare, the
reflection spectrum for a normal DBR (no-chirping and $\Lambda=400~{}nm$) is
also shown in 3(d). It is evident that there is $\approx~{}240~{}nm$ increase
in PBG for C-DBR as compared to the normal DBR. Also, the reflectivity drop at
the band-edges is relatively smooth with complete suppression of small
reflection resonances outside the PBG. In fact, any alteration in structural
parameters has an impact $\Delta\beta$ and $\kappa$ and consequently, the
adiabatic constraints are not satisfied closely. This is expected to reduce
the PBG in reflection spectrum along with appearance of sharp transmission
resonance (outside PBG). For example, when the chirp-length changes to
$\delta=5~{}nm$ with $d_{1}=100~{}nm$ and $N=39$, the reflection spectrum is
shown in Fig. 3(e) where the PBG shrinks to $\approx~{}182~{}nm$. It is also
worth noting that the reflection spectrum is accompanied by oscillating side-
bands with sharp transmission resonances on both ends of the PBG. On reducing
the chirp length further to $\delta=2.5~{}nm$ ($d_{1}=150~{}nm$ and $N=39$),
PBG for the C-DBR shrinks to $100~{}nm$ and discernibly sharper transmission
resonances on both ends of PBG which is similar to that exhibited by normal
DBR. The underlying cause behind this observation could be traced to the
variation of $\kappa$ (see equation (16)). By reducing the chirp length
(without changing the DBR length $L$), the minimum value of $\kappa$ (at
$z=0$) and $z=L$) increases. Therefore, the forward and backward propagating
modes are not completely decoupled at the ends of C-DBR when $\delta=5~{}nm$
and $\delta=2.5~{}nm$. However, reducing the chirp has a weak impact on
$\frac{d\Delta\beta}{dz}$. Overall, adiabaticity conditions are partially
satisfied for smaller chirp length and consequently, we obtain a smaller PBG.

Figure 4: Shows the evolution of state vector $\vec{S}$ on the Bloch sphere
for the normal DBR at wavelengths a) $\lambda_{c}=1650~{}nm$ (central
wavelength) b) $\lambda=1500~{}nm$ c) $\lambda=1000~{}nm$ d)
$\lambda=2500~{}nm$ and C-DBR at wavelengths (e) $\lambda_{c}=1650~{}nm$
(central wavelength) (f) $\lambda=1500~{}nm$ (g) $\lambda=1000~{}nm$ and (h)
$\lambda=2500~{}nm$ respectively.

.

###  2.3 Geometric representation of propagation characteristics in DBR

The analogy between a two-level atomic system and propagation dynamics in a
DBR is apparent from Eqs. (12) and (13) where the time-evolution has been
replaced by evolution along $z$-axis. The ground-state and excited state
populations are analogues to complex amplitudes $\tilde{a_{i}}$ and
$\tilde{a_{r}}$ respectively. The parameters determining the geometrical path
$\Delta$ (frequency detuning) and $\Omega$ (Rabi frequency) are replaced by
$\Delta k$ (phase-mismatch) and $\kappa$ (coupling coefficient). Therefore,
the dynamical trajectory on the Bloch-sphere for a perfectly phase-matched
($\Delta k=0$) interaction in the DBR should resemble an on-resonance
interaction in two-level atomic system. Consequently, a $\pi$-pulse
($\kappa.z=\pi$) at the resonance is equivalent to exactly satisfying the
Bragg’s condition which manifests into complete conversion of $\tilde{a_{i}}$
into $\tilde{a_{r}}$ resulting in reflectivity $R=1$. On an equivalent Bloch-
sphere, Fig. 4(a) represent such an evolution where the state-vector
($\vec{S}$) traverses from south-pole to the north-pole along the great circle
(red-curve). Here, the DBR has periodicity $\Lambda=400~{}nm$,
$d_{1}=d_{2}=\frac{\Lambda}{2}$ and the PBG central wavelength is
$\lambda_{c}=1650~{}nm$ (where $\Delta k=0$). Any non-zero $\Delta k$
(analogous to detuned interaction) would mean a smaller mode-conversion. In
case of frequencies within the PBG (say $\lambda=1500~{}nm$), the dynamical
trajectory would result in termination of the state-vector at some point on
the surface of northern-hemisphere of the DBR Bloch sphere as shown in Fig.
4(b). Interestingly, for frequencies outside the PBG ($|\Delta k|>>0$) or in
the transmission band (say $\lambda=1000~{}nm$ or $\lambda=2500~{}nm$), the
dynamical trajectory traced by $\vec{S}$ terminates within the southern
hemisphere which could be observed in Figs. 4(c) and (d) respectively. The
conversion efficiency ($\eta=\frac{S_{z}+1}{2}$) as a function of propagating
distance ($z$) for all the aforementioned frequencies in normal DBR is shown
in Fig.5(a). It could be noted that $\eta<1$ for all wavelengths except
central PBG wavelength ($\lambda_{c}=1650~{}nm$) which is consistent with the
description in Fig. 4.

The geometric representation for a C-DBR is shown in Fig. 4 for frequencies
within the PBG (Fig. 4e,f) as well as outside the PBG (Fig. 4g,h). Since, the
parameters $\Delta k$ and $\kappa$ exhibit a _slow_ longitudinal variation,
$\vec{S}$ always remain perpendicular to $\vec{B}$ at each $z$. Therefore,
$\vec{S}$ follows a spiralling trajectory after beginning its journey from the
south-pole. Figure 4(e) represents a dynamical trajectory of $\vec{S}$ (at
$\lambda_{c}=1650~{}nm$) for the variation in $\Delta k$ (or $\Delta\beta$)
shown in Fig. 3(a). In this case, $\vec{S}$ traverses from the south-pole to
the north-pole through a spiralling trajectory and this results in complete
complete conversion of $\ket{i}$ to $\ket{r}$ as shown in Fig. 5(b). For
another wavelength $\lambda=1500~{}nm$ which is within the PBG, the dynamical
trajectory appears similar but it exhibits asymmetrically distributed spirals
(with respect to the equator) while travelling from the south-pole to the
north-pole of DBR Bloch sphere (see Fig. 4(b). In this case, the conversion
efficiency reaches unity through a different path as shown in Fig. 5(b). On
the other hand, $\vec{S}$ remains within the southern hemisphere for
wavelength outside the PBG (see Figs. 4(g) and (h)).

Figure 5: a) and b) shows the variation of conversion efficiency as a function
of propagation distance for normal DBR and C-DBR at wavelengths used in Fig.
4. The black solid line represents the evolution of the central wavelength of
the PBG, the green solid lines for wavelength that lie closer to the band-
edges. The blue/purple solid lines correspond to wavelengths that lie outside
the PBG

.

Figure 6: a) and b) shows the variation reflection spectrum for TE and TM
polarization respectively in a normal DBR ($\Lambda=400~{}nm$ and
$d_{1}=d_{2}$) as a function of angle of incidence. c) and d) shows the
reflection spectrum for TE and TM polarization respectively in C- DBR as a
function of angle of incidence. The C-DBR parameters are $\delta=~{}10~{}nm$
and $d_{1}=10~{}nm$. All the cases, total number of units is $N=39$

.

###  2.4 Oblique incidence and angular dispersion

A detailed comparison of reflection spectrum and dispersion for a normal DBR
and a C-DBR (with $\delta=10~{}nm$) is elucidated in Fig. 6(a)-(d). The
dependence of reflection spectrum on angle-of-incidence (AOI) for TE and TM
polarization in a normal-DBR is shown in Fig. 6(a) and (b) respectively. The
oblique incidence essentially leads to a smaller value of normal component of
the wavevector and consequently, the PBGs shift to higher frequencies with
increasing angle of incidence. As expected, the PBGs tend to broaden for the
TE polarization on oblique incidence. On the other hand, the low frequency
PBGs for TM polarization tends to reduce up to an AOI = $\approx 60^{\circ}$
and increases thereafter. The drop in PBG for TM polarization is essentially
due to the Brewster’s effect at the interface of high and low-index layers.
Figure 6(c) and (d) represent the reflection spectrum for the TE and TM
polarization respectively for C-DBR. A comparison between Fig. 6(a) and (c)
reveals that the C-DBR exhibits appreciably broad PBGs with very narrow
transmission bands separating them. At oblique incidence, the PBG in C-DBR
broadens further and shifts to higher frequencies. This is accompanied by
shrinking of transmission bands. In fact, the PBGs tend to overlap for AOI
$\geq~{}50^{\circ}$, thereby leading to a high-reflection band extending from
$150~{}THz$ to $750~{}THz$ ($1600~{}nm$ band). Such broad PBGs are usually not
found in normal DBR, even with very wide refractive index contrast. It is also
interesting to note that the reflection spectrum (in Fig. 6(c)) exhibits three
omnidirectional PBGs which are located in $160-250~{}THz$, $310-400~{}THz$ and
$490-550~{}THz$ frequency range. The PBG, in this case (for normal as well as
oblique incidence) is limited by the material transparency window and could be
extended further (on both spectral ends) through suitable choice of materials.
A similar comparison between Fig. 6(b) and (d) depict broadened PBG in case of
TM polarization in C-DBR and narrow transmission bands. However, Fig. 6(d)
exhibits a sharp transmission resonance at the Brewster’s angle
($\approx~{}60^{\circ}$). The PBGs tend to merge thereafter
($\geq~{}65^{\circ}$) giving rise to a broad high reflection band. A
comparison between Fig. 6(c) and (d)reveals that the C-DBR could be employed
as a broadband $(\approx 150-750~{}THz)$ polarization filter for
$\approx~{}60^{\circ}$ AOI.

##  3 Conclusion

In conclusion, we present an approach to understand the propagation
characteristics of modes in a DBR using general techniques adopted in a wide
variety of systems exhibiting _SU(2)_ dynamical symmetry. The coupled-mode
equations describing the forward and backward propagating modes in a DBR could
be represented in the form of a single _optical_ Bloch-equation where the
evolution of state-vector depicts the dynamical behaviour of the wave
propagation. This provides a platform to draw an analogy with a _two-level_
atomic system and consequently, adopt a formalism for adiabatic evolution in
DBR based configurations. In order to realize conditions imposed by adiabatic
constraints, C-DBR configurations have been investigated in detail. The DBR
variants exhibit enhancement of PBG along with varying degree of suppression
of sharp transmission resonances in the reflection spectrum. The impact of
alteration in physical parameters of the DBR is explored in detail. It is
worth pointing out that the C-DBR configuration involves a discernible
longitudinal variation in mode-coupling coefficient $\kappa$ in addition to
the sweep in $\Delta k$ (or $\Delta\beta$). Conventionally, such adiabatic
process have a close resemblance with the _Allen-Eberly_ scheme defined in the
context of population-transfer in _two-level_ atomic systems[19]. Novel DBR
designs could further be explored which satisfy the adiabatic constraints. An
interesting extension of this proposal would be to investigate the evolution
of _geometric_ -phase in such DBR configurations and the possibility to
control the backscattered (reflection) phase through suitable DBR design. This
promises to provide an unique and flexible platform for tailoring the spatial
features of an optical beam using adiabatic DBR configurations. Nevertheless,
a natural extension of this proposal would be to explore the viability of this
formalism in _two_ \- and _three_ -dimensional photonic crystals with focus on
applications such as sensing and enhanced nonlinear optical interactions.

##  4 Disclosures

The authors declare that there are no conflicts of interest related to this
article.

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