# Positive Knots and Ribbon Concordance

Joe Boninger  Department of Mathematics, Boston College, Chestnut Hill, MA
<EMAIL_ADDRESS>
###### Abstract.

Ribbon concordances between knots generalize the notion of ribbon knots. Agol,
building on work of Gordon, proved ribbon concordance gives a partial order on
knots in $S^{3}$. In previous work, the author and Greene conjectured that
positive knots are minimal in this ordering. In this note we prove this
conjecture for a large class of positive knots, and show that a positive knot
cannot be expressed as a non-trivial band sum—both results extend earlier
theorems of Greene and the author for special alternating knots. In a related
direction, we prove that if positive knots $K$ and $K^{\prime}$ are concordant
and $|\sigma(K)|\geq 2g(K)-2$, then $K$ and $K^{\prime}$ have isomorphic
rational Alexander modules. This strengthens a result of Stoimenow, and gives
evidence toward a conjecture that any concordance class contains at most one
positive knot.

##  1\. Introduction

A smooth concordance between knots $K_{0},K_{1}\subset S^{3}$ is a smooth,
properly embedded cylinder $C\subset S^{3}\times I$ such that
$C\cap(S^{3}\times\\{i\\})=K_{i}$ for $i=0,1$. Perturbing $C$ if necessary, we
assume the height function $h:C\hookrightarrow S^{3}\times I\to I$ is Morse,
and we say $C$ is a ribbon concordance from $K_{1}$ to $K_{0}$ if $h$ has no
critical points of index two. If such a concordance exists, we say $K_{1}$ is
ribbon concordant to $K_{0}$ and we write $K_{0}\leq K_{1}$. This terminology
generalizes the notion of a ribbon knot, since a knot is ribbon if and only if
it is ribbon concordant to the unknot.

In a now-classic paper, Gordon conjectured that ribbon concordance induces a
partial ordering on the set of knots [14]. This conjecture was settled in the
affirmative by Agol [1], and many authors have shown that ribbon concordance
places strong constraints on knot invariants. To give just a few examples, if
$K_{0}\leq K_{1}$, then:

  * •

The Alexander polynomial $\Delta_{K_{0}}$ of $K_{0}$ divides $\Delta_{K_{1}}$
[13, 11].

  * •

The genus $g(K_{0})$ of $K_{0}$ is less than or equal to $g(K_{1})$ [39].

  * •

If $K_{1}$ is fibered, then $K_{0}$ is as well [23, 39].

Gordon also proved torus knots are minimal under ribbon concordance [14].
Torus knots are examples of positive knots, knots admitting a diagram in which
all crossings are positive. This motivated the author and Josh Greene to
conjecture:

######  Conjecture 1.1 ([6, Conjecture 1.6],[38, Question 1.3]).

If $K_{1}\subset S^{3}$ is a positive knot and $K_{0}\leq K_{1}$, then
$K_{0}\cong K_{1}$.

1.1 was also posed independently by Tagami, who proved it for positive two-
bridge knots [38]. The author and Greene proved the conjecture for fibered
positive knots, following a similar result of Baker and Motegi [5].

In this note we prove two theorems in support of 1.1. First, given a two-
component split link $K_{0}\sqcup K_{1}\subset S^{3}$ and an embedded band
$b=I\times I\subset S^{3}$ satisfying $b\cap K_{i}=I\times\\{i\\}$ for
$i=0,1$, we define the band sum $K_{0}\\#_{b}K_{1}\subset S^{3}$ by

$K_{0}\\#_{b}K_{1}=(K_{0}\cup K_{1}-I\times\partial I)\cup\partial I\times I.$

This band sum is trivial if there exists a sphere $\Sigma\subset
S^{3}-(K_{0}\cup K_{1})$ which intersects $b$ in a single arc—in this case
$K_{0}\\#_{b}K_{1}\cong K_{0}\\#K_{1}$, the ordinary connect sum. We say a
knot is band prime if it cannot be written as a non-trivial band sum.

Miyazaki proved any band sum $K_{0}\\#_{b}K_{1}$ is ribbon concordant to the
connect sum $K_{0}\\#K_{1}$ [22]. Thus, band sums are a natural way in which
ribbon concordances arise. Additionally, if $K^{\prime}\leq K$ via a ribbon
concordance with only two critical points, then basic Morse theory shows $K$
is equivalent to a band sum of $K^{\prime}$ and an unknot. We prove:

######  Theorem 1.2.

Positive knots are band prime.

We also verify 1.1 for a large class of positive knots.

######  Theorem 1.3.

Let $K\subset S^{3}$ be a positive knot. If the leading coefficient of
$\Delta_{K}$ is a prime power, then $K$ is ribbon concordance minimal.

We list 1.3 second because its proof is somewhat simpler than that of 1.2. In
[6], the author and Greene proved Theorems 1.2 and 1.3 for all special
alternating knots; since special alternating knots are positive (up to
mirroring), Theorems 1.2 and 1.3 are broad extensions of those results.
Additionally, while the proof that special alternating knots are band prime in
[6] has a combinatorial flavor, our proof of 1.2 is more geometric. We use a
theorem of Ozawa stating any incompressible Seifert surface of a positive knot
is free [25], and we show that such a Seifert surface cannot witness a non-
trivial, genus-preserving band sum.

In a related direction, we consider the following question of Gordon:

######  Question 1.4 ([14, Question 6.1]).

Does every smooth concordance class contain a unique representative which is
minimal with respect to ribbon concordance?

Affirmative answers to this question and to [14, Question 6.2] would imply a
generalization of the Slice-Ribbon Conjecture. 1.4 also seems closely related
to conjectures made independently by other authors: Rudolph conjectured that
each concordance class contains at most one algebraic knot [29] and Baker
conjectured each concordance class contains at most one fibered knot
supporting the tight contact structure [4]. Most relevant to us, Stoimenow
conjectured each concordance class contains finitely many positive knots
[35]—this was verified by Baader, Dehornoy and Liechti [3]. Considering 1.1
and 1.4, it is natural to posit:

######  Conjecture 1.5 ([35]).

Every smooth concordance class contains at most one positive knot.

We credit Stoimenow since 1.5 seems implicit in his work. For any knot $K$,
let $d(K)$ denote the degree of $\Delta_{K}$ when normalized to have no
negative exponents. As evidence of 1.5, we prove:

######  Theorem 1.6.

Let $K$ and $K^{\prime}$ be (topologically or algebraically) concordant
positive knots. If $K$ satisfies $|\sigma(K)|\geq d(K)-2$, where $\sigma$
denotes the signature, then the rational Alexander modules of $K$ and
$K^{\prime}$ are isomorphic.

By the rational Alexander module of $K$ we mean the cohomology ring
$H^{*}(\bar{X};\mathbb{Q})$, where $\bar{X}$ denotes the infinite cyclic cover
of the exterior of $K$, viewed as a module over the group of deck
transformations. 1.6 strengthens a result of Stoimenow, which concluded under
the above hypotheses that $K$ and $K^{\prime}$ have the same Alexander
polynomial [35, Theorem 4.5]. Additionally, although the hypothesis that

(1) $|\sigma(K)|\geq d(K)-2$

is somewhat restrictive, it is known that the signatures of positive knots are
linearly bounded from below by their genus [3]. In fact, (1) holds for all
positive knots with genus less than or equal to four with the single exception
of the knot $14_{45657}$ [35, Theorem 2.4][8, 34, 36].

######  Corollary 1.7.

Let $K$ and $K^{\prime}$ be (topologically or algebraically) concordant
positive knots. If $g(K)\leq 4$, then the rational Alexander modules of $K$
and $K^{\prime}$ are isomorphic.

Condition (1) also includes all special alternating knots, since these satisfy
$|\sigma(K)|=d(K)$. We prove 1.6 by showing that positive knots which satisfy
(1) are $\mathbb{Q}$-anisotropic—for a definition of $\mathbb{Q}$-anisotropy,
see Section 5 below. By a classical result of Kervaire and Gilmer,
algebraically concordant knots which are $\mathbb{Q}$-anisotropic and admit
non-singular Seifert matrices have isomorphic rational Alexander modules [13,
Proposition 4.2] [17]. Thus, we could remove the requirement (1) from 1.6 if
we knew that:

######  Conjecture 1.8.

Positive knots are $\mathbb{Q}$-anisotropic.

By work of Gilmer, 1.8 may be thought of as the statement that positive knots
are algebraically ribbon concordance minimal [13, Theorem 0.1]. Scharlemann
proved 1.8 for the case of torus knots [33, Proposition 2.3].

###  1.1. Further discussion

Section 5 below contains some results on roots of Alexander polynomials which
may be of independent interest—for example, we show that Alexander polynomials
of positive knots have no rational roots. We also remark that our proof of 1.6
extends to almost positive knots, knots which admit a diagram with one
negative crossing, using results of Tagami and Stoimenow [37][35, Theorem
2.3]. It may be interesting to consider whether Theorems 1.2 and 1.3 could
also be extended to almost positive knots.

###  1.2. Outline

In Section 2 we recall relevant properties of postiive knots, in Section 3 we
prove 1.2, and in Section 4 we prove 1.3. In Section 5 we discuss
$\mathbb{Q}$-anisotropy and prove 1.6 and 1.7.

###  1.3. Acknowledgements

The author thanks Josh Greene for guidance with math and writing, and Ian
Biringer for a helpful conversation about handlebodies. This material is based
upon work supported by the National Science Foundation under Award No.
2202704.

##  2\. Properties of Positive Knots

We gather some useful facts about positive knots. First, by results of
Rudolph, the genus $g(K)$ and slice genus $g_{4}(K)$ of a positive knot $K$
are equal [30, 31]. This motivates the following lemma, well known to experts:

######  Lemma 2.1.

Let $K_{0},K_{1}\subset S^{3}$ be such that $K_{0}\leq K_{1}$ and $K_{1}$
satisfies $g(K_{1})=g_{4}(K_{1})$. Then $g(K_{0})=g(K_{1})$.

In particular, the conclusion of Lemma 2.1 holds if $K_{0}\leq K_{1}$ and
$K_{1}$ is positive.

###### Proof.

Since genus is non-increasing under ribbon concordance [39] and slice genus is
a concordance invariant, we have

$g_{4}(K_{1})=g_{4}(K_{0})\leq g(K_{0})\leq g(K_{1})=g_{4}(K_{1}).$

∎

Given Rudolph’s work, it is natural to ask whether minimal genus Seifert
surfaces of positive knots are special in some way. A result in this direction
was proven by Ozawa. A Seifert surface $S\subset S^{3}$ is called free if
$S^{3}-\nu(S)$ is a handlebody, where $\nu$ denotes a regular open
neighborhood; equivalently, $S$ is free if $\pi_{1}(S^{3}-S)$ is a free group.

######  Theorem 2.2 ([25, Corollary 1.2]).

If $K$ is a positive knot, then every incompressible Seifert surface of $K$ is
free.

Finally, we will use the fact that positive knots are pseudo-alternating (not
be confused with quasi-alternating!). The precise definition of pseudo-
alternating will not be important to us (see [19, Section 4]), but for experts
we note that pseudo-alternating links are those links which can be built from
Murasugi sums of special alternating links. Positive knots are pseudo-
alternating because they are homogeneous [9].

##  3\. Positive Knots are Band Prime

In this section we prove 1.2, first recalling some standard definitions from
three-manifold topology. Let $Y$ be a three-manifold and $\Sigma\subset Y$ a
properly embedded surface. A compressing disk for $\Sigma$ is an embedded disk
$D\subset Y$ with $D\cap\Sigma=\partial D$, such that $\partial D$ does not
bound a disk in $\Sigma$. Similarly, a boundary-compressing disk for $\Sigma$
is a disk $D\subset Y$ such that:

  * •

$D\cap\Sigma\subset\partial D$

  * •

$\partial D$ consists of an arc in $\partial Y$ and an arc in $\Sigma$ which
is not boundary-parallel in $\Sigma$.

The arc $\partial D\cap\Sigma$ is called a boundary-compressing arc. The
surface $\Sigma$ is called compressible (resp. boundary-compressible) if it
admits a compressing disk (resp. boundary-compressing disk), and
incompressible (resp. boundary-incompressible) otherwise.

We will need a couple lemmas on surfaces in handlebodies—the first is a
classical fact.

######  Lemma 3.1 ([15, Example III.13]).

Let $H$ be a handlebody. If $\Sigma\subset H$ is a connected surface which is
incompressible and boundary-incompressible, then $\Sigma$ is a disk.

In the next lemma, by a planar surface we mean a compact surface which can be
embedded in $\mathbb{R}^{2}$.

######  Lemma 3.2.

Let $H$ be a handlebody with boundary $F=\partial H$. Let $\Sigma\subset H$ be
a properly embedded, connected planar surface such that:

  * •

$\Sigma$ is incompressible in $H$, and

  * •

The components of $\partial\Sigma$ are separating and parallel to one another
in $F$.

Then $\Sigma$ is either a disk or a boundary-parallel annulus.

###### Proof.

We will suppose $\Sigma$ has more than one boundary component (i.e. that
$\Sigma$ is not a disk) and show $\Sigma$ is a boundary-parallel annulus. By
Lemma 3.1, since $\Sigma$ is incompressible it is boundary-compressible. Let
$D$ be a boundary-compressing disk for $\Sigma$, so that $\partial
D=\alpha\cup\alpha^{\prime}$ with $\alpha^{\prime}\subset F$ and $\alpha$ a
properly embedded arc in $\Sigma$ which is not boundary-parallel.

We first suppose the two boundary points
$\partial\alpha=\partial\alpha^{\prime}$ lie in distinct boundary components
$\gamma$ and $\gamma^{\prime}$ of $\partial\Sigma$. Since $\gamma$ and
$\gamma^{\prime}$ are parallel in $F$ they cobound an annulus $A\subset F$,
and since they are separating $\alpha^{\prime}$ lies in $A$. Let $N$ be a
regular neighborhood of $A\cup D$ in $H-\Sigma$, and let

$B=N\cap\Sigma=\partial N\cap\Sigma.$

Equivalently, $B$ is a regular neighborhood of
$\gamma\cup\alpha\cup\gamma^{\prime}$ in $\Sigma$. Let $C=\partial N-(A\cup
B)$, as in Figure 1.

2pt $A$ at 75 370 $B$ at 18 315 $C$ at 133 12 $D$ at 400 405 $\gamma$ at 40 55
$\gamma^{\prime}$ at 502 193 $\alpha$ at 435 325

Figure 1.

The neighborhood $N$ is a thickened annulus, so $\partial N$ is a torus. Since
$A\cup B$ is a torus with a disk removed, as the figure shows, it follows that
$C$ is a disk. Now

$\partial C=\partial B\subset\Sigma,$

and the incompressibility of $\Sigma$ implies $\partial C$ bounds a disk
$C^{\prime}$ in $\Sigma$. Thus $\Sigma=B\cup C^{\prime}$, and in this case
$\Sigma$ is an annulus with boundary $\gamma\cup\gamma^{\prime}$. The disks
$C$ and $C^{\prime}$ are parallel since they cobound a three-ball, and it
follows that $\Sigma$ is boundary-parallel.

We have shown that if $\Sigma$ is not an annulus, then for any boundary-
compressing disk $D$ with $\partial D=\alpha\cup\alpha^{\prime}$ as above, the
endpoints of $\alpha\subset\Sigma$ lie in the same component of
$\partial\Sigma$. The planarity of $\Sigma$ then implies each such $\alpha$
separates $\Sigma$. We therefore choose a boundary-compressing disk $D$ whose
compressing arc $\alpha\subset\Sigma$ is “outermost,” i.e. one of the two
components of $\Sigma-\alpha$ does not contain any boundary-compressing arcs
which are not isotopic to $\alpha$. Let $\Sigma^{\prime}$ denote this
component of $\Sigma-\alpha$.

Finally, we consider the surface
$\Sigma^{\prime\prime}=\Sigma^{\prime}\cup_{\alpha}D$. Since $D$ is a
boundary-compressing disk $\alpha$ is not boundary-parallel in $\Sigma$, so
neither $\Sigma^{\prime}$ nor $\Sigma^{\prime\prime}$ is a disk. Additionally,
$\Sigma^{\prime\prime}$ is incompressible: suppose, toward a contradiction,
that $\Sigma^{\prime\prime}$ admits a compressing disk $D^{\prime}$. Then
$\partial D^{\prime}$ may be isotoped into $\Sigma^{\prime}$, and the
incompressibility of $\Sigma$ implies $\partial D^{\prime}$ bounds a disk in
$\Sigma$. Thus $\Sigma$ is the union of $\Sigma^{\prime}$ with a disk, which
contradicts the assumption that $\alpha$ is not boundary-parallel in $\Sigma$.
A similar argument, using the fact that $\alpha$ is outermost in $\Sigma$,
shows $\Sigma^{\prime\prime}$ is boundary-incompressible. This contradicts
Lemma 3.1, and we conclude that $\Sigma$ is an annulus as before. ∎

1.2 now follows from the more general 3.3 below. Positive knots satisfy the
hypotheses of 3.3 by the discussion in Section 2.

######  Theorem 3.3.

Let $K\subset S^{3}$ be a knot such that:

  * •

$g(K)=g_{4}(K)$, and

  * •

Every minimal genus Seifert surface of $K$ is free.

Then $K$ is band prime.

###### Proof.

Let $K$ be a knot satisfying the given hypotheses, and suppose $K$ can be
written as a band sum of knots $K_{0},K_{1}\subset S^{3}$. We will show $K$ is
equivalent to the standard connect sum $K_{0}\\#K_{1}$; by an observation of
Miyazaki, this occurs if and only if the band sum is trivial [24] (cf. [10]).

As mentioned in the introduction, Miyazaki also showed $K$ is ribbon
concordant to the connect sum $K_{0}\\#K_{1}$ [22]. Thus, by Lemma 2.1,

$g(K)=g(K_{0}\\#K_{1})=g(K_{0})+g(K_{1}).$

Gabai proved that a band sum preserves genus in the above sense if and only if
there exists a minimal genus Seifert surface $S$ for $K$, such that $S$ is a
band sum

$S=S_{0}\\#_{b}S_{1}$

of Seifert surfaces $S_{i}$ for $K_{i}$, $i=0,1$ [12]. In other words, $S$ is
the result of joining the split union $S_{0}\sqcup S_{1}$ along a band $b$.
The band $b$ may be different from the band in our initial band sum
representation of $K$, but this is allowed by the discussion in the first
paragraph. We fix such a surface $S$, and let $\Sigma\subset S^{3}$ be a
sphere separating the component surfaces $S_{0}$ and $S_{1}$. We choose
$\Sigma$ transverse to $S$ so that the number of intersection components
$|\Sigma\cap S|$ is minimal among all such spheres; then $\Sigma\cap S$
consists of a set of parallel co-cores of $b$, and $|\Sigma\cap S|$ is odd
since $\Sigma$ separates the feet of $b$.

Let $\nu(S)$ be a regular neighborhood of $S$, and let $H=S^{3}-\nu(S)$. We
claim the planar surface

$\Sigma\cap H=\Sigma-\nu(S)$

is incompressible in $H$. Suppose not: then $\Sigma\cap H$ admits a
compressing disk $D$. The curve $\partial D$ separates $\Sigma$ into
components $\Sigma_{1}$ and $\Sigma_{2}$, and each component $\Sigma_{i}$
contains some intersection with $S$ since $\partial D$ is essential in
$\Sigma-\nu(S)$. Because $\Sigma$ separates $S_{0}$ and $S_{1}$, one of the
spheres $\Sigma_{0}\cup D$ or $\Sigma_{1}\cup D$ does as well. Assuming the
former without loss of generality, we conclude that $\Sigma_{0}\cup D$ is a
sphere separating $S_{0}$ and $S_{1}$ with

$|(\Sigma_{0}\cup D)\cap S|<|\Sigma\cap S|.$

This contradicts the minimality of $|\Sigma\cap S|$, proving the claim.

The surface $S$ has minimal genus, so $H$ is a handlebody by hypothesis. Now
$\Sigma\cap H$ is an incompressible planar surface in $H$, and since
$\Sigma\cap S$ consists of a set of parallel co-cores of $b$,
$\partial(\Sigma\cap H)$ consists of a set of parallel curves which are
separating on $\partial H$. From Lemma 3.2, since $|\Sigma\cap S|$ is odd, we
conclude that $|\Sigma\cap S|=1$ and $\Sigma-S$ is a disk. Thus the band sum
$K_{0}\\#_{b}K_{1}$ is trivial and $K$ is band prime. ∎

######  Remark 3.4.

It is not true that strongly quasi-positive knots are band prime—see [5,
Section 4.1]—but it is true that fibered strongly quasi-positive knots are
band prime by a result of Baker and Motegi [5, Theorem 1.1]. In fact Baker and
Motegi’s argument shows that fibered strongly quasi-positive knots, like
fibered positive knots, are ribbon concordance minimal.

##  4\. A Condition for Minimality

In this section we prove 1.3, which involves piecing together several existing
results. First, the lower central series $\\{\gamma_{i}\\}_{i\geq 0}$ of a
group $G$ is defined recursively by

$\displaystyle\gamma_{0}$ $\displaystyle=G$ $\displaystyle\gamma_{i}$
$\displaystyle=[\gamma_{i-1},G]\text{ for all }i>0,$

where $[*,*]$ indicates the commutator. The group $G$ is residually nilpotent
if $\bigcap_{i=0}^{\infty}\gamma_{i}=\\{1\\}$, and following Gordon [14] we
say a knot $K\subset S^{3}$ is residually nilpotent if the commutator subgroup
of the knot group is residually nilpotent. As in the introduction, let $d(K)$
denote the degree of $\Delta_{K}$. Then Gordon proves:

######  Lemma 4.1 ([14, Lemma 3.4]).

Let $K_{0},K_{1}\subset S^{3}$ be knots with $K_{0}\leq K_{1}$. If $K_{1}$ is
residually nilpotent and $d(K_{0})=d(K_{1})$, then $K_{0}\cong K_{1}$.

Fibered knots are examples of residually nilpotent knots, since their
commutator subgroups are free (see [18, Chapter 5]), but little is known in
general about which knots are residually nilpotent. Murasugi and Mayland
proved the following theorem:

######  Theorem 4.2 ([19]).

Let $K$ be a pseudo-alternating knot such that the leading coefficient of
$\Delta_{K}$ is a prime power. Then $K$ is residually nilpotent.

Next, we recall some background on knot Floer homology [27, 28]. The hat
version of knot Floer homology, $\widehat{HFK}$, associates a finitely
generated, bigraded $\mathbb{F}_{2}$-vector space to any knot $K$:

$\widehat{HFK}(K)=\bigoplus_{i,j\in\mathbb{Z}}\widehat{HFK}_{i}(K,j).$

The $i$ and $j$ gradings are called the Maslov and Alexander gradings
respectively, and the graded Euler characteristic of $\widehat{HFK}$ is the
symmetrized Alexander polynomial:

(2) $\Delta_{K}(t)=\sum_{i,j}(-1)^{i}\dim(\widehat{HFK}_{i}(K,j))t^{j}.$

Generalizing the classical fact that $d(K)\leq 2g(K)$, knot Floer homology
detects knot genus in the following sense [26]:

$g(K)=\max\\{j\mid\widehat{HFK}(K,j)\neq 0\\}.$

Any concordance $C\subset S^{3}\times I$ between knots $K_{0}\subset
S^{3}\times\\{0\\}$ and $K_{1}\subset S^{3}\times\\{1\\}$ induces a bigrading-
preserving homomorphism

$C_{*}:\widehat{HFK}(K_{0})\to\widehat{HFK}(K_{1}),$

and Zemke proved that if $C$ is a ribbon concordance from $K_{1}$ to $K_{0}$,
so $K_{0}\leq K_{1}$, then the map $C_{*}$ is injective [39]. Finally, we
require the following theorem of Cheng, Hedden and Sarkar:

######  Theorem 4.3 ([7, Corollary 1.6]).

If $K$ is a pseudo-alternating link, then the top Alexander grading
$\widehat{HFK}(K,g(K))$ of $\widehat{HFK}(K)$ is supported in a single Maslov
grading.

1.3 now follows easily from these results and the next proposition (cf. [6,
Propositon 1.4]).

######  Proposition 4.4.

Let $K_{1}$ be a pseudo-alternating knot such that $g(K)=g_{4}(K)$, and
suppose $K_{0}\leq K_{1}$. Then $\Delta_{K_{0}}=\Delta_{K_{1}}$.

###### Proof.

By Lemma 2.1, $g(K_{0})=g(K_{1})=g$ for some $g\in\mathbb{N}$. Fix a ribbon
concordance $C\subset S^{3}\times I$ from $K_{1}$ to $K_{0}$, and let $C_{*}$
denote the induced map

$C_{*}:\widehat{HFK}(K_{0},g)\to\widehat{HFK}(K_{1},g).$

This map is injective by Zemke’s result, and both groups are nonzero since the
knots have the same genus. By 4.3 $\widehat{HFK}(K_{1};g)$ is supported in a
single Maslov grading, and thus $\widehat{HFK}(K_{0};g)$ is as well.
Consequently, (2) implies that

$d(K_{0})=2g(K_{0})=2g(K_{1})=d(K_{1}).$

Since $\Delta_{K_{0}}$ divides $\Delta_{K_{1}}$, we have
$\Delta_{K_{0}}=m\Delta_{K_{1}}$ for some $m\in\mathbb{Z}$ [13]. But

$\Delta_{K_{0}}(1)=\Delta_{K_{1}}(1)=1,$

so $\Delta_{K_{0}}=\Delta_{K_{1}}$. ∎

###### Proof of 1.3..

Suppose $K_{0}$ and $K_{1}$ are knots such that $K_{1}$ satisfies the
hypotheses of the theorem and $K_{0}\leq K_{1}$. By 4.4,
$\Delta_{K_{0}}=\Delta_{K_{1}}$. Additionally $K_{1}$ is residually nilpotent
by 4.2, so Lemma 4.1 implies $K_{1}\cong K_{0}$. ∎

Two-bridge knots are residually nilpotent by a theorem of Johnson [16]. Thus
our proof of 1.3 also gives an alternate proof of Tagami’s theorem that
positive two-bridge knots are ribbon concordance minimal, using [16, Corollary
1.3] in place of 4.2.

##  5\. Alexander Modules and $\mathbb{Q}$-Anisotropy

We now work toward the proofs of 1.6 and 1.7. We expect the following
proposition is known to experts, but we have not been able to find it in the
literature.

######  Proposition 5.1.

Let $K\subset S^{3}$ be a knot such that $\Delta_{K}$ has a rational root $q$.
Then $q=(a-1)/a$ for some integer $a\notin\\{0,1\\}$. In particular, $q$ is
positive.

###### Proof.

Let $q=a/b$ for $a,b\in\mathbb{Z}$. Fix an oriented Seifert surface $S$ for
$K$, and let

$\iota_{\pm}:H_{1}(S)\to H_{1}(S^{3}-S)$

be the maps induced by pushing curves off $S$ to the $\pm$-component of the
unit normal bundle of $S$, with sign determined by the orientation. Since
Alexander duality yields a canonical isomorphism $H_{1}(S^{3}-S)\cong
H_{1}(S)$, it makes sense to discuss the determinants of $\iota_{+}$ and
$\iota_{-}$.

Now

$\Delta_{K}(t)=\det(\iota_{+}-t\iota_{-}),$

and $\Delta_{K}(q)=0$ implies that

$0=\det(b\iota_{+}-a\iota_{-}).$

Therefore there exists a non-zero vector $v\in H_{1}(S)$ such that

$b\iota_{+}(v)=a\iota_{-}(v).$

Dividing by a scalar if necessary, we assume $v$ is primitive, i.e. that $v$
extends to a basis of $H_{1}(S)$.

The intersection pairing

$\cdot:H_{1}(S)\times H_{1}(S)\to\mathbb{Z}$

satisfies the identity

$v_{1}\cdot v_{2}=\text{lk}(v_{1},(\iota_{+}-\iota_{-})(v_{2}))$

for all $v_{1},v_{2}\in H_{1}(S)$, where lk indicates the linking number.
Since $v$ is a primitive homology class on a once-punctured surface, $v$ is
representable by a simple, closed non-separating curve on $S$ [32, 20], and it
follows that there exists $w\in H_{1}(S)$ such that $w\cdot v=1$.

We have

$1=\text{lk}(w,(\iota_{+}-\iota_{-})(v))=\text{lk}(w,\iota_{+}(v))-\text{lk}(w,\iota_{-}(v))$

and therefore

$a\text{lk}(w,\iota_{+}(v))=b\text{lk}(w,\iota_{-}(v))=b(\text{lk}(w,\iota_{+}(v))-1).$

Since $a$ and $b$ are both non-zero, lk$(w,\iota_{+}(v))\notin\\{0,1\\}$. We
conclude that

$q=\frac{a}{b}=\frac{\text{lk}(w,\iota_{+}(v))-1}{\text{lk}(w,\iota_{+}(v))}$

as desired. ∎

######  Corollary 5.2.

If $K$ is a positive knot, then $\Delta_{K}$ has no rational roots.

###### Proof.

Let $K$ be positive. The Conway polynomial of $K$,
$\nabla_{K}(z)\in\mathbb{Z}[z^{2}]$, is the unique polynomial satisfying

$\nabla_{K}(x-x^{-1})=\Delta_{K}(x^{2})$

where $\Delta_{K}$ is the symmetrized Alexander polynomial. Cromwell proved
that for a positive knot $K$, the coefficients of $\nabla_{K}$ are non-
negative [9, Corollary 2.1]. Furthermore

$\nabla_{K}(0)=\Delta_{K}(1)=1,$

so $\nabla_{K}$ has no real roots. It follows that $\Delta_{K}$ has no
positive real roots, since a positive real root $q$ of $\Delta_{K}$ would
yield a real root $\sqrt{q}-1/\sqrt{q}$ of $\nabla_{K}$. Therefore, by 5.1,
$\Delta_{K}$ has no rational roots. ∎

Let $\bar{X}$ denote the infinite cyclic cover of the exterior of a knot $K$.
For any field $\mathbb{F}$, an invariant $\mathbb{F}$-isotropic subspace of
$H^{1}(\bar{X};\mathbb{F})\cong H^{1}(\bar{X},\partial\bar{X};\mathbb{F})$ is
one which is preserved by the action of deck transformations and self-
annihilating with respect to the cup product

$\smile:H^{1}(\bar{X};\mathbb{F})\times
H^{1}(\bar{X},\partial\bar{X};\mathbb{F})\to
H^{2}(\bar{X},\partial\bar{X};\mathbb{F})\cong\mathbb{F}.$

The knot $K$ is called $\mathbb{F}$-anisotropic if $H^{1}(\bar{X};\mathbb{F})$
does not contain a non-trivial invariant $\mathbb{F}$-isotropic subspace. As
the introduction discusses, $\mathbb{Q}$-anisotropy can be used to restrict
changes to the Alexander module under concordance: Kervaire and Gilmer proved
that if (algebraically) concordant knots $K$ and $K^{\prime}$ are
$\mathbb{Q}$-anisotropic and admit Seifert matrices which are invertible over
$\mathbb{Q}$, then the rational Alexander modules of $K$ and $K^{\prime}$ are
isomorphic [13, Proposition 4.2] [17].

######  Proposition 5.3.

If a positive knot $K$ satisfies $|\sigma(K)|\geq d(K)-2$, then $K$ is
$\mathbb{Q}$-anisotropic.

###### Proof.

Let $\bar{X}$ denote the infinite cyclic cover of the exterior of $K$, and let

$t:H^{1}(\bar{X};\mathbb{Q})\to H^{1}(\bar{X};\mathbb{Q})$

be the map induced by a primitive deck transformation. Let $\Lambda\subset
H^{1}(\bar{X};\mathbb{Q})$ be a non-trivial invariant $\mathbb{Q}$-isotropic
subspace of $H^{1}(\bar{X};\mathbb{Q})$.

Up to a scalar, the characteristic polynomial of $t$ coincides with
$\Delta_{K}$. Since the cup product is skew-symmetric, it is straightforward
to check that $H^{1}(\bar{X};\mathbb{Q})$ contains a one-dimensional invariant
$\mathbb{Q}$-isotropic subspace if and only if $\Delta_{K}$ has a rational
root, i.e. if and only if $t$ has a rational eigenvalue (cf. [14, Proposition
4.3]). Thus, 5.2 implies dim$(\Lambda)\geq 2$.

We now consider the Milnor form [21] $\mu$ on $H^{1}(\bar{X};\mathbb{Q})$,
defined by

$\mu(v,w)=t(v)\smile w+t(w)\smile v$

for $v,w\in H^{1}(\bar{X};\mathbb{Q})$. As Gordon observes and is easy to
check, $\Lambda$ is also a self-annihilating subspace for $\mu$ [14,
Proposition 4.5]. Let $V_{\pm}$ denote a maximal subspace of
$H^{1}(\bar{X};\mathbb{Q})$ on which $\mu$ is $\pm$-definite. Then
$V_{\pm}\cap\Lambda=\\{0\\}$, so

$\dim V_{\pm}\leq\dim H^{1}(\bar{X};\mathbb{Q})-\dim\Lambda\leq\dim
H^{1}(\bar{X};\mathbb{Q})-2=d(K)-2.$

It follows that

$|\sigma(K)|=|\sigma(\mu)|\leq d(K)-4,$

and since $d(K)$ and $\sigma(K)$ are even this implies the desired inequality.
∎

######  Remark 5.4.

It is not true that positive knots are $\mathbb{R}$-anisotropic, even if they
satisfy the hypothesis of 5.3: for example, the Alexander polynomial of the
positive knot $10_{139}$ has a negative real root. It is also not true that
strongly quasi-positive knots are $\mathbb{Q}$-anistropic, since there exist
strongly quasi-positive knots which are topologically slice—see, for example,
[2].

###### Proof of 1.6.

Let $K$ and $K^{\prime}$ be concordant positive knots such that
$|\sigma(K)|\geq d(K)-2$. Then $\sigma(K)=\sigma(K^{\prime})$, and since $K$
and $K^{\prime}$ are positive we have

$d(K)=g_{4}(K)=g_{4}(K^{\prime})=d(K^{\prime}).$

It follows that $|\sigma(K^{\prime})|\geq d(K^{\prime})-2$, so $K$ and
$K^{\prime}$ are both $\mathbb{Q}$-anisotropic by 5.3. Since $g(K)=d(K)$ and
$g(K^{\prime})=d(K^{\prime})$, any Seifert matrix of a minimal genus Seifert
surface for $K$ or $K^{\prime}$ is invertible over $\mathbb{Q}$. Thus, by the
result of Kervaire and Gilmer discussed before 5.3, the rational Alexander
modules of $K$ and $K^{\prime}$ are isomorphic. ∎

###### Proof of 1.7.

Since the knot $14_{45657}$ is only the positive knot satisfying $g(K)=4$ and
$\sigma(K)=-4$, $14_{45657}$ is not concordant to any other positive knot. The
corollary then follows from 1.6 and the discussion in the introduction. ∎

## References

  * [1] Ian Agol, _Ribbon concordance of knots is a partial ordering_ , Comm. Amer. Math. Soc. 2 (2022), 374–379.
  * [2] S. Baader, P. Feller, L. Lewark, and L. Liechti, _On the topological 4-genus of torus knots_ , Trans. Amer. Math. Soc. 370 (2018), no. 4, 2639–2656.
  * [3] Sebastian Baader, Pierre Dehornoy, and Livio Liechti, _Signature and concordance of positive knots_ , Bull. Lond. Math. Soc. 50 (2018), no. 1, 166–173.
  * [4] Kenneth L. Baker, _A note on the concordance of fibered knots_ , J. Topol. 9 (2016), no. 1, 1–4.
  * [5] Kenneth L. Baker and Kimihiko Motegi, _Tight fibered knots and band sums_ , Math. Z. 286 (2017), no. 3-4, 1357–1365.
  * [6] Joe Boninger and Joshua Evan Greene, _Special Alternating Knots Are Band Prime_ , International Mathematics Research Notices (2024), rnae009.
  * [7] Zhechi Cheng, Matthew Hedden, and Sucharit Sarkar, _Murasugi sum and extremal knot floer homology_ , 2022.
  * [8] Tim D. Cochran and Robert E. Gompf, _Applications of Donaldson’s theorems to classical knot concordance, homology $3$-spheres and property $P$_, Topology 27 (1988), no. 4, 495–512.
  * [9] P. R. Cromwell, _Homogeneous links_ , J. London Math. Soc. (2) 39 (1989), no. 3, 535–552.
  * [10] Mario Eudave-Muñoz, _Band sums of links which yield composite links. the cabling conjecture for strongly invertible knots_ , Trans. Amer. Math. Soc. 330 (1992), no. 2, 463–501.
  * [11] Stefan Friedl and Mark Powell, _Homotopy ribbon concordance and Alexander polynomials_ , Arch. Math. (Basel) 115 (2020), no. 6, 717–725.
  * [12] David Gabai, _Genus is superadditive under band connected sum_ , Topology 26 (1987), no. 2, 209–210.
  * [13] Patrick M. Gilmer, _Ribbon concordance and a partial order on $S$-equivalence classes_, Topology Appl. 18 (1984), no. 2-3, 313–324.
  * [14] C. McA. Gordon, _Ribbon concordance of knots in the $3$-sphere_, Math. Ann. 257 (1981), no. 2, 157–170.
  * [15] William Jaco, _Lectures on three-manifold topology_ , CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, RI, 1980.
  * [16] Jonathan Johnson, _Residual torsion-free nilpotence, bi-orderability and two-bridge links_ , 2021.
  * [17] Michel A. Kervaire, _Knot cobordism in codimension two_ , Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Math., vol. Vol. 197, Springer, Berlin-New York, 1971, pp. 83–105.
  * [18] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, _Combinatorial group theory_ , second ed., Dover Publications, Inc., Mineola, NY, 2004.
  * [19] E. J. Mayland, Jr. and Kunio Murasugi, _On a structural property of the groups of alternating links_ , Canadian J. Math. 28 (1976), no. 3, 568–588.
  * [20] Mark D. Meyerson, _Representing homology classes of closed orientable surfaces_ , Proc. Amer. Math. Soc. 61 (1976), no. 1, 181–182 (1977).
  * [21] John W. Milnor, _Infinite cyclic coverings_ , Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 115–133.
  * [22] Katura Miyazaki, _Band-sums are ribbon concordant to the connected sum_ , Proc. Amer. Math. Soc. 126 (1998), no. 11, 3401–3406.
  * [23] Katura Miyazaki, _A note on genera of band sums that are fibered_ , J. Knot Theory Ramifications 27 (2018), no. 12, 1871002, 3.
  * [24] by same author, _When is a band-connected sum equal to the connected sum?_ , Topology Appl. 272 (2020), 107071, 5.
  * [25] Makoto Ozawa, _Closed incompressible surfaces in the complements of positive knots_ , Comment. Math. Helv. 77 (2002), no. 2, 235–243.
  * [26] Peter Ozsváth and Zoltán Szabó, _Holomorphic disks and genus bounds_ , Geom. Topol. 8 (2004), 311–334.
  * [27] by same author, _Holomorphic disks and knot invariants_ , Adv. Math. 186 (2004), no. 1, 58–116.
  * [28] Jacob Andrew Rasmussen, _Floer homology and knot complements_ , ProQuest LLC, Ann Arbor, MI, 2003, Thesis (Ph.D.)–Harvard University.
  * [29] Lee Rudolph, _How independent are the knot-cobordism classes of links of plane curve singularities?_ , Notices Amer. Math. Soc. 23 (1976), 410\.
  * [30] by same author, _Quasipositivity as an obstruction to sliceness_ , Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51–59.
  * [31] by same author, _Positive links are strongly quasipositive_ , Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 555–562.
  * [32] James A. Schafer, _Representing homology classes on surfaces_ , Canad. Math. Bull. 19 (1976), no. 3, 373–374.
  * [33] Martin Scharlemann, _The fundamental group of fibered knot cobordisms_ , Math. Ann. 225 (1977), no. 3, 243–251.
  * [34] A. Stoimenow, _Gaußdiagram sums on almost positive knots_ , Compos. Math. 140 (2004), no. 1, 228–254.
  * [35] by same author, _Application of braiding sequences III: Concordance of positive knots_ , Internat. J. Math. 26 (2015), no. 7, 1550050, 36.
  * [36] Alexander Stoimenow, _Diagram genus, generators, and applications_ , Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016\.
  * [37] Keiji Tagami, _The Rasmussen invariant, four-genus and three-genus of an almost positive knot are equal_ , Canad. Math. Bull. 57 (2014), no. 2, 431–438.
  * [38] Keiji Tagami, _Remarks on the minimalities of two-bridge knots in the ribbon concordance poset_ , Bulletin of the Belgian Mathematical Society - Simon Stevin 30 (2023), no. 3, 317 – 327.
  * [39] Ian Zemke, _Knot Floer homology obstructs ribbon concordance_ , Ann. of Math. (2) 190 (2019), no. 3, 931–947.


# IMS-Speech: A Speech to Text Tool

Pavel Denisov, Ngoc Thang Vu

###### Abstract

We present the IMS-Speech, a web based tool for German and English speech
transcription aiming to facilitate research in various disciplines which
require accesses to lexical information in spoken language materials. This
tool is based on modern open source software stack, advanced speech
recognition methods and public data resources and is freely available for
academic researchers. The utilized models are built to be generic in order to
provide transcriptions of competitive accuracy on a diverse set of tasks and
conditions.

##  1 Introduction

There is a considerable amount of spoken language materials in form of audio
recordings, which researchers in e.g. humanities and social science could
incorporate into their studies. However, to be able to access to their
content, one needs to automatically transcribe these recordings. While all
needed resources for building of an automatic speech recognition (ASR) system
are typically available for academic usage, their utilization requires
specialized knowledge and technical experience [1], [2]. Therefore, in order
to provide people easy accesses to information in spoken language materials, a
speech to text tool with a user interface should be helpful.

This paper presents the IMS-Speech111http://www.ims.uni-
stuttgart.de/forschung/ressourcen/werkzeuge/IMS-Speech.html, a web based tool
for German and English speech transcription aiming to facilitate research in
various disciplines. We are willing to provide a speech transcription service
with an intuitive web interface accessible with a wide range of computing
devices and to people with various backgrounds. The service is based on modern
open source software stack, advanced speech recognition methods and public
data resources and is freely available for academic researchers. The utilized
models are built to be generic in order to provide transcriptions of
competitive accuracy on a diverse set of tasks and conditions. In addition to
that, they can serve as a strong base for customized task specific
applications.

##  2 System description

In order to produce a meaningful transcription for the most of recordings that
might be uploaded by users, two tasks must be performed for every recording
sequentially. First, a recording must be split to segments not exceeding some
short duration and corresponding to speech intervals. Second, actual ASR must
be performed over each speech segment for finding the most probable sequence
of words being said in the segment and thus constructing final transcription.

###  2.1 Speech Segmentation

Speech segmentation is performed with a speech activity detection (SAD) system
based on Time-Delay Neural Network (TDNN) [3] with statistics pooling [4] for
long-context information. TDNN is trained to estimate probability of 3
classes, Silence, Speech and Garbage, for each frame. Training targets are
assigned based on lattices produced by Gaussian Mixture Model (GMM) based
acoustic models and predefined lists of phones for each class. GMM is used for
forced alignment as well as for unconstrained decoding. Training targets are
obtained from both procedures separately and consequently merged by weighted
summing, while samples with high disagreement between two methods are
discarded. During the decoding, 3 estimated probabilities are transformed to
pseudo-likelihoods of 2 states, Silence and Speech, using priors of 3 classes
and manually chosen proportions of 2 states in 3 classes. Decoding is
performed with Viterbi algorithm [5].

###  2.2 End-to-end ASR

End-to-end approach implements ASR system as a single neural network based
model that takes a $T$-length sequence of $d$ dimensional feature vectors
$X=\\{x_{t}\in\mathbb{R}^{d}|t=1,\dots,T\\}$ in the input and provides a
$U$-length sequence of output labels
$Y=\\{y_{u}\in\mathcal{U}|u=1,\dots,U\\}$, where $\mathcal{U}$ is a set of
distinct output labels and usually $U<T$. Common architecture for such models
is attention-based encoder-decoder network trained to minimize cross-entropy
loss:

$\displaystyle\mathcal{L}_{\text{att}}=$
$\displaystyle-\log\mathnormal{p}_{\text{att}}(Y|X)$ (1)
$\displaystyle\mathnormal{p}_{\text{att}}(Y|X)=$
$\displaystyle\prod_{u}\mathnormal{p}(y_{u}|X,y_{1:u-1})$ (2)
$\displaystyle\mathnormal{p}(y_{u}|X,y_{1:u-1})=$ $\displaystyle\text{
Decoder}(\mathbf{r}_{u},\mathbf{q}_{u-1},y_{u-1})$ (3)
$\displaystyle\mathbf{h}_{t}=$ $\displaystyle\text{ Encoder}(X)$ (4)
$\displaystyle a_{ut}=$ $\displaystyle\text{
Attention}(\\{a_{u-1}\\}_{t},\mathbf{q}_{u-1},\mathbf{h}_{t})$ (5)
$\displaystyle\mathbf{r}_{u}=$ $\displaystyle\sum_{t}a_{ut}\mathbf{h}_{t}.$
(6)

Here, $\text{Encoder}(\cdot)$ and $\text{Decoder}(\cdot)$ are recurrent neural
networks, $\text{Attention}(\cdot)$ is an attention mechanism and
$\mathbf{h}_{t}$, $\mathbf{q}_{u-1}$ and $\mathbf{r}_{u}$ are the hidden
vectors. Attention mechanism has been developed in the context of machine
translation problem [6] and provides a means to model correspondence of all
elements of hidden representations sequence to all elements of output sequence
in the decoder. Attention mechanism allows to learn non-sequential mapping
between its inputs and outputs, meaning that order of output elements is not
always the same as order of input elements corresponding to them, what can be
an advantage in case of machine translation task, as word order sometimes
differs between languages. However, this property of attention mechanism makes
training of speech recognition suboptimal, because it is known in advance that
word order is the same in audio and in transcription. Connectionist Temporal
Classification (CTC) sequence level loss function [7] has been adopted as a
secondary learning objective for end-to-end ASR models in order to suppress
this drawback:

$\displaystyle\mathcal{L}=\lambda\mathcal{L}_{\text{ctc}}+(1-\lambda)\mathcal{L}_{\text{att}},$
(7)

where $0\leq\lambda\leq 1$. Encoder output followed by a single linear layer
serves as estimated output label sequence in CTC loss calculation, while
target is set to be all possible $T$-length sequences of an extended output
labels set $Z=\\{z_{t}\in\mathcal{U}\cup\texttt{<blank>}|t=1,\dots,T\\}$,
corresponding to the original output labels sequence $Y$:

$\displaystyle\mathcal{L}_{\text{ctc}}=$
$\displaystyle-\log\mathnormal{p}_{\text{ctc}}(Y|X)$ (8)
$\displaystyle\mathnormal{p}_{\text{ctc}}(Y|X)\triangleq$
$\displaystyle\sum_{Z}\prod_{t}\mathnormal{p}(z_{t}|z_{t-1},Y)\mathnormal{p}(z_{t}|X)$
(9) $\displaystyle\mathnormal{p}(z_{t}|X)=$ $\displaystyle\text{
Softmax}(\text{Lin}(\mathbf{h}_{t})).$ (10)

It has been found that CTC output can also improve decoding results when
combined with the main attention-based probabilities during the search:

$\displaystyle\hat{Y}=\operatorname*{arg\,max}_{Y}\\{\lambda\log\mathnormal{p}_{\text{ctc}}(Y|X)+(1-\lambda)\log\mathnormal{p}_{\text{att}}(Y|X)\\}.$
(11)

External language model (LM) is commonly-used technique to improve ASR
results. LMs are trained on text corpora, which usually contain order of
magnitude more examples of written language compared to acoustic corpora, and
therefore provide a reliable source of information for selection of well
formed transcriptions from hypotheses. In end-to-end ASR, this information is
used during the decoding by adding LM probability of hypothetical output label
sequence with scaling factor $\gamma$ to probabilities obtained from the main
model:

$\displaystyle\hat{Y}=\operatorname*{arg\,max}_{Y}\\{\lambda\log\mathnormal{p}_{\text{ctc}}(Y|X)+(1-\lambda)\log\mathnormal{p}_{\text{att}}(Y|X)+\gamma\log\mathnormal{p}_{\text{lm}}(Y)\\}.$
(12)

Encoder-decoder architecture allows output sequence (transcription) to have
any length that does not exceed length of input sequence (audio recording).
Consequently, it is possible to employ different kinds of output units, for
example words or characters. In case of words, transcription hypotheses are
limited by words presenting in vocabulary, what causes out of vocabulary
problems and requires large dimensionality of final layers. In case of
characters, output sequences become very long for alphabetical languages, what
leads to high number of hypothetical transcriptions and slows down the
decoding. Sub-word units have been suggested first as a trade-off solution in
machine translation [8] and recently have been adopted in speech recognition
[9]. Sub-word units include single characters and can be used to encode any
word. In addition to that, sub-word units include combinations of several
characters and encode words to shorter sequences compared to single
characters. Unigram language model algorithm [10] performs segmentation of a
string $X$ by searching for the most probable sequence of sub-word units
composing the string:

$\displaystyle\mathbf{x}^{*}=\operatorname*{arg\,max}_{\mathbf{x}\in\mathcal{S}(X)}P(\mathbf{x}),$
(13)

where probability $P(\mathbf{x})$ of a sequence of sub-word units
$\mathbf{x}=(x_{1},\ldots,x_{M})$ is defined as the product of occurrence
probabilities of sub-word units:

$\displaystyle P(\mathbf{x})=\prod_{i=1}^{M}p(x_{i}),$ (14)
$\displaystyle\forall
i\,\,x_{i}\in\mathcal{V},\,\,\,\sum_{x\in\mathcal{V}}p(x)=1.$

Sub-word units vocabulary $\mathcal{V}$ is derived during the training of
segmentation model by starting from some large set of frequent in the training
data substrings and iterative elimination of certain percent of substrings
having lowest impact on total likelihood of all possible sequences of sub-word
units for all sentences until some predefined size of vocabulary is reached.

##  3 Implementation

The frontend is implemented as a Node.js/React application and utilizes
WebSocket protocol to communicate with the backend. Users can sign in and
upload their recordings for transcription. We plan to add the users’ feedback
with the main focus on customization and fine tuning.

Speech segmentation is performed with Kaldi toolkit [1]. We use the pretrained
SAD model downloaded from http://kaldi-asr.org/models/m4. The model is trained
on Fisher-English corpus [11] augmented with room impulses and additive noise
from Room Impulse Response and Noise Database [12]. The input features of SAD
model are 40-dimensional Mel Frequency Cepstral Coefficients (MFCC) without
cepstral truncation with a frame length 25 ms and shift of 10 ms. We use the
segmentation parameters suggested in aspire Kaldi recipe, but extend maximum
speech segment duration from 10 to 30 seconds and enable consecutive speech
segments merging when duration of merged segment does not exceed 10 seconds.

Speech recognition is implemented with ESPnet end-to-end speech recognition
toolkit [2] with PyTorch backend. We follow LibriSpeech ESPnet recipe and use
80-dimensional log Mel filterbank coefficients concatenated with 3-dimensional
pitch having a frame length of 25 ms and shift of 10 ms as acoustic features
and sub-word units as output labels. Kaldi toolkit is used to extract and
normalize input features. Normalization to zero mean and unit variance is done
with global statistics from the training data. SentencePiece unsupervised text
tokenizer222https://github.com/google/sentencepiece is used to generate list
of sub-word units based on the language model training data and to segment all
kinds of text data. We evaluated several sizes of sub-word unit vocabulary
between 50 and 5000 and found that 100 resulted in better results for both
English and German systems. The ASR model is an encoder-decoder neural
network. The encoder network consists of 2 VGG [13] blocks followed by 5
Bidirectional Long Short-Term Memory Network (BLSTM) layers [14] with 1024
units in each layer and direction. The decoder network consists of 2 Long
Short-Term Memory Network (LSTM) [15] layers with 1024 units and location
based attention mechanism with 1024 dimensions, 10 convolution channels and
100 convolution filters. CTC weight $\lambda$ is set to $0.5$ for both
training and decoding. Training is performed with AdaDelta optimizer [16] and
gradient clipping on 4 Graphics Processing Units (GPUs) in parallel with a
batch size of 24 for 10 epochs. The optimizer is initialized with $\rho=0.95$
and $\epsilon=10^{-8}$. $\epsilon$ is halved after an epoch if performance of
the model did not improve on validation set. The model with the highest
accuracy on validation set is used for the decoding with beam size of 20.

External LM for the English system contains 2 layers of 650 LSTM units and is
trained with stochastic gradient descent optimizer with batch size 256 for 60
epochs. LM scaling factor $\gamma$ is set to $0.5$ during decoding for the
English system. External LM for the German system contains 2 layers of 3000
LSTM units and is trained with Adam optimizer [17] with batch size 128 for 10
epochs. LM scaling factor $\gamma$ is set to $1.1$ during decoding for the
German system.

##  4 Resources

Both English and German systems are trained on multiple speech databases,
which are summarized in Table 1. We use data preparation scripts from multi_en
Kaldi recipe and German ASR recipe [18]. German system is additionally
improved by data augmentation, applied to 3 datasets (marked with (*) in the
table) with Acoustic Simulator333https://github.com/idiap/acoustic-simulator
package. This procedure gives an augmented dataset that is 10 times larger
than original dataset.

External LM for the English system is trained with on transcriptions from the
training speech databases except of Common Voice. External LM for the German
system is trained on all transcriptions form the training speech databases and
additional text corpus444http://ltdata1.informatik.uni-
hamburg.de/kaldi_tuda_de/German_sentences_8mil_filtered_maryfied.txt.gz
containing 8 millions of preprocessed read sentences from the German
Wikipedia, the European Parliament Proceedings Parallel Corpus and a crawled
corpus of direct speech.

Table 1: English and German training data covering data sets with different styles Language | Corpus | Style | Hours
---|---|---|---
English |  LibriSpeech [19] | Read | 960
Switchboard [20] | Spontaneous | 317
TED-LIUM 3 [21] | Spontaneous | 450
AMI [22] | Spontaneous | 229
WSJ [23] | Read | 81
Common Voice555https://voice.mozilla.org/en/datasets | Read | 240
Total |  | 2277
German |  Tuda-De [24] | Read | 109
SWC [25] | Read | 245
M-AILABS666http://www.m-ailabs.bayern/en/the-mailabs-speech-dataset/ (*) | Read | 2336
Verbmobil 1 and 2 [26] (*) | Mixed | 417
VoxForge777http://www.voxforge.org/de/Downloads (*) | Read | 571
RVG 1 [27] | Mixed | 100
PhonDat 1 [28] | Mixed | 19
Total |  | 3797

##  5 ASR Performance

###  5.1 Results

Table 2 compares the results of IMS-Speech on several testing datasets with
the best results for the corresponding datasets which we could find in various
sources. In summary, these results suggest that our generic systems can
compete with task specific systems and in some cases even outperform them,
possibly due to better generalization from larger amount of training data.

Table 2: ASR performance comparison with state of the art results (WER, %) Language | Dataset | IMS-Speech | State of the art
---|---|---|---
English | WSJ eval’92 | 3.8 |  3.5 [29]
LibriSpeech test-clean | 4.4 |  3.2 [30]
LibriSpeech test-other | 12.7 |  7.6 [30]
TED-LIUM 3 test | 12.8 |  6.7 [21]
AMI IHM eval | 17.4 |  19.2888https://github.com/kaldi-asr/kaldi/blob/4bdb05ae78a842a07cae326aeb32aea87328fb2c/egs/ami/s5b/RESULTS_ihm#L87
AMI SDM eval | 38.5 |  36.7999https://github.com/kaldi-asr/kaldi/blob/4bdb05ae78a842a07cae326aeb32aea87328fb2c/egs/ami/s5b/RESULTS_sdm#L105
AMI MDM eval | 34.1 |  34.2101010https://github.com/kaldi-asr/kaldi/blob/4bdb05ae78a842a07cae326aeb32aea87328fb2c/egs/ami/s5b/RESULTS_mdm#L97
German | Tuda-De dev | 11.1 |  13.1 [18]
Tuda-De test | 12.0 |  14.4 [18]
Verbmobil 1 dev | 6.7 |  18.2 [18]
Verbmobil 1 test | 7.3 |  12.7 [31]

We evaluate the recognition speech with different beam widths and batched
recognition with inference using CPU and GPU. The results in Table 3 show that
batched recognition can significantly increase speed of recognition without
any impact on WER.

Table 3: Beam width effect on recognition performance and speed on Tuda-De test set Beam width |  Inference on 1 CPU core with batch size of 1 |  Inference on 1 GPU with batch size of 23
---|---|---
WER, % | RT factor | WER, % | RT factor
20 | 12.0 | 14.2 | 12.0 | 0.7
15 | 12.2 | 11.3 | 12.2 | 0.5
10 | 12.6 | 8.8 | 12.6 | 0.4
5 | 13.7 | 7.0 | 13.7 | 0.3

###  5.2 Comparisions with Google API

We use the ASR benchmark framework [32] to compare performance of IMS-Speech
and Google API. The results of Google API were retrieved on 8.01.2019. As the
framework uses custom WER computation method instead of NIST sclite utility
used in ESPnet recipes, we had to perform scoring of IMS-Speech output with
the framework as well. We excluded all utterances for which Google API
transcriptions contained digits, because WER would be high for them even if
transcriptions were correct (a couple of examples are given in Table 4), and
also utterances for which Google API transcriptions were empty. The results
are shown in Table 5. The numbers suggest that that Google API models may be
optimized for certain speech domain and recording conditions that differ
significantly from the ones tested by us.

Table 4: Examples of some perfect IMS-Speech transcriptions and Google API transcriptions Utterance | System | Transcription
---|---|---
LibriSpeech test-other, 2609-157645-0010 | IMS-Speech |  then let them sing to the hundred and nineteenth replied the curate
Google API |  then let them sing the 119th repository
Verbmobil 1 test, w007dxx0_001_BFG | IMS-Speech |  Ich würde Ihnen den einundzwanzigsten August bis zum vier fünfundzwanzigsten vorschlagen
Google API |  ich würde Ihnen den 21. August bis den 425 vorschlagen
Table 5: ASR performance comparison with Google API in term of WER (%) Language | Dataset | IMS-Speech | Google API | Scored utterances
---|---|---|---|---
English | LibriSpeech test-clean | 4.3 | 15.9 | 2444 of 2620 (93%)
LibriSpeech test-other | 12.5 | 28.0 | 2708 of 2939 (92%)
Common Voice valid-test | 4.5 | 19.2 | 3772 of 3995 (94%)
German | Tuda-De test | 10.0 | 12.4 | 3481 of 4100 (85%)
Verbmobil 1 test | 8.7 | 19.5 | 334 of 631 (53%)

##  6 Conclusion

We presented IMS-Speech, a web based speech transcription tool for English and
German languages that can be used by non-technical researchers in order to
utilize the information from audio recordings in their studies. The comparison
of the IMS-Speech results with the results of specialized systems in terms of
WER showed that the described service can perform decently in a diverse set of
tasks and conditions. In the future, we plan to allow the users to customize
the system for their needs as well as to constantly improve our ASR system.

## References

  * Povey et al. [2011] Povey, D., A. Ghoshal, G. Boulianne, L. Burget, O. Glembek, N. Goel, M. Hannemann, P. Motlicek, Y. Qian, P. Schwarz et al.: _The Kaldi speech recognition toolkit_.  In _Proc. of ASRU_. 2011.
  * Watanabe et al. [2018] Watanabe, S., T. Hori, S. Karita, T. Hayashi, J. Nishitoba, Y. Unno, N. E. Y. Soplin, J. Heymann, M. Wiesner, N. Chen et al.: _ESPnet: End-to-End Speech Processing Toolkit_.  _arXiv preprint arXiv:1804.00015_ , 2018.
  * Waibel et al. [1990] Waibel, A., T. Hanazawa, G. Hinton, K. Shikano, and K. J. Lang: _Phoneme recognition using time-delay neural networks_.  In _Readings in speech recognition_. 1990.
  * Ghahremani et al. [2016] Ghahremani, P., V. Manohar, D. Povey, and S. Khudanpur: _Acoustic Modelling from the Signal Domain Using CNNs_.  In _Proc. of Interspeech_. 2016.
  * Viterbi [1967] Viterbi, A.: _Error bounds for convolutional codes and an asymptotically optimum decoding algorithm_.  _IEEE Transactions on Information Theory_ , 1967.
  * Bahdanau et al. [2014] Bahdanau, D., K. Cho, and Y. Bengio: _Neural machine translation by jointly learning to align and translate_.  _arXiv preprint arXiv:1409.0473_ , 2014.
  * Graves et al. [2006] Graves, A., S. Fernández, F. Gomez, and J. Schmidhuber: _Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks_.  In _Proc. of ICML_. 2006.
  * Sennrich et al. [2015] Sennrich, R., B. Haddow, and A. Birch: _Neural machine translation of rare words with subword units_.  _arXiv preprint arXiv:1508.07909_ , 2015.
  * Zeyer et al. [2018] Zeyer, A., K. Irie, R. Schlüter, and H. Ney: _Improved training of end-to-end attention models for speech recognition_.  _arXiv preprint arXiv:1805.03294_ , 2018.
  * Kudo [2018] Kudo, T.: _Subword Regularization: Improving Neural Network Translation Models with Multiple Subword Candidates_.  _arXiv preprint arXiv:1804.10959_ , 2018.
  * Cieri et al. [2004] Cieri, C., D. Miller, and K. Walker: _The Fisher Corpus: a Resource for the Next Generations of Speech-to-Text_.  In _LREC_. 2004.
  * Ko et al. [2017] Ko, T., V. Peddinti, D. Povey, M. L. Seltzer, and S. Khudanpur: _A study on data augmentation of reverberant speech for robust speech recognition_.  In _Proc. of IEEE ICASSP_. 2017.
  * Simonyan and Zisserman [2014] Simonyan, K. and A. Zisserman: _Very deep convolutional networks for large-scale image recognition_.  _arXiv preprint arXiv:1409.1556_ , 2014.
  * Graves et al. [2005] Graves, A., S. Fernández, and J. Schmidhuber: _Bidirectional LSTM networks for improved phoneme classification and recognition_.  In _International Conference on Artificial Neural Networks_ , pp. 799–804. Springer, 2005.
  * Hochreiter and Schmidhuber [1997] Hochreiter, S. and J. Schmidhuber: _Long short-term memory_.  _Neural computation_ , 9(8), pp. 1735–1780, 1997.
  * Zeiler [2012] Zeiler, M. D.: _ADADELTA: an adaptive learning rate method_.  _arXiv preprint arXiv:1212.5701_ , 2012.
  * Kingma and Ba [2014] Kingma, D. P. and J. Ba: _Adam: A method for stochastic optimization_.  _arXiv preprint arXiv:1412.6980_ , 2014.
  * Milde and Köhn [2018] Milde, B. and A. Köhn: _Open Source Automatic Speech Recognition for German_.  In _Proc. of ITG_. 2018.
  * Panayotov et al. [2015] Panayotov, V., G. Chen, D. Povey, and S. Khudanpur: _Librispeech: an ASR corpus based on public domain audio books_.  In _Proc. of IEEE ICASSP_. 2015.
  * Godfrey et al. [1992] Godfrey, J. J., E. C. Holliman, and J. McDaniel: _SWITCHBOARD: Telephone speech corpus for research and development_.  In _Proc. of IEEE ICASSP_. 1992.
  * Hernandez et al. [2018] Hernandez, F., V. Nguyen, S. Ghannay, N. Tomashenko, and Y. Esteve: _TED-LIUM 3: twice as much data and corpus repartition for experiments on speaker adaptation_.  _arXiv preprint arXiv:1805.04699_ , 2018.
  * Carletta [2007] Carletta, J.: _Unleashing the killer corpus: experiences in creating the multi-everything AMI Meeting Corpus_.  _Language Resources and Evaluation_ , 2007.
  * Paul and Baker [1992] Paul, D. B. and J. M. Baker: _The design for the Wall Street Journal-based CSR corpus_.  In _Proc. of the workshop on Speech and Natural Language_. 1992.
  * Radeck-Arneth et al. [2015] Radeck-Arneth, S., B. Milde, A. Lange, E. Gouvêa, S. Radomski, M. Mühlhäuser, and C. Biemann: _Open source german distant speech recognition: Corpus and acoustic model_.  In _Text, Speech, and Dialogue_. 2015.
  * Köhn et al. [2016] Köhn, A., F. Stegen, and T. Baumann: _Mining the Spoken Wikipedia for Speech Data and Beyond_.  In _Proc. of LREC_. 2016.
  * Wahlster [2013] Wahlster, W.: _Verbmobil: foundations of speech-to-speech translation_.  Springer Science & Business Media, 2013.
  * Burger and Schiel [1998] Burger, S. and F. Schiel: _RVG 1 – A Database for Regional Variants of Contemporary German_.  In _Proc. of LREC_. 1998.
  * Hess et al. [1995] Hess, W. J., K. J. Kohler, and H.-G. Tillmann: _The Phondat-verbmobil speech corpus_.  In _Fourth European Conference on Speech Communication and Technology_. 1995.
  * Chan and Lane [2015] Chan, W. and I. Lane: _Deep recurrent neural networks for acoustic modelling_.  _arXiv preprint arXiv:1504.01482_ , 2015.
  * Han et al. [2017] Han, K. J., A. Chandrashekaran, J. Kim, and I. Lane: _The CAPIO 2017 conversational speech recognition system_.  _arXiv preprint arXiv:1801.00059_ , 2017.
  * Gaida et al. [2014] Gaida, C., P. Lange, R. Petrick, P. Proba, A. Malatawy, and D. Suendermann-Oeft: _Comparing open-source speech recognition toolkits_.  _Tech. Rep., DHBW Stuttgart_ , 2014.
  * Dernoncourt et al. [2018] Dernoncourt, F., T. Bui, and W. Chang: _A Framework for Speech Recognition Benchmarking_.  _Proc. of Interspeech_ , 2018.


CI

    confidence interval
OLS

    ordinary least squares
CLT

    central limit theorem
IV

    instrumental variables
ATE

    average treatment effect
RCT

    randomized control trial
VAM

    value-added model
LAN

    locally asymptotically normal
DiD

    difference-in-differences
OVB

    omitted variables bias
FWL

    Frisch-Waugh-Lovell

# Excess death rates for Republicans and Democrats during the COVID-19
pandemic††thanks: We gratefully acknowledge support from the Tobin Center for
Economic Policy at Yale University and the Yale School of Public Health
COVID-19 Rapid Response Research Fund.

Jacob Wallace Yale School of Public Health Email<EMAIL_ADDRESS>Paul
Goldsmith-Pinkham Yale School of Management and NBER Email: paul.goldsmith-
<EMAIL_ADDRESS>Jason L. Schwartz Yale School of Public Health Email:
<EMAIL_ADDRESS>
0.1cm0.1cm

Political affiliation has emerged as a potential risk factor for COVID-19,
amid evidence that Republican-leaning counties have had higher COVID-19 death
rates than Democrat-leaning counties and evidence of a link between political
party affiliation and vaccination views. This study constructs an individual-
level dataset with political affiliation and excess death rates during the
COVID-19 pandemic via a linkage of 2017 voter registration in Ohio and Florida
to mortality data from 2018 to 2021. We estimate substantially higher excess
death rates for registered Republicans when compared to registered Democrats,
with almost all of the difference concentrated in the period after vaccines
were widely available in our study states. Overall, the excess death rate for
Republicans was 5.4 percentage points (pp), or 76%, higher than the excess
death rate for Democrats. Post-vaccines, the excess death rate gap between
Republicans and Democrats widened from 1.6 pp (22% of the Democrat excess
death rate) to 10.4 pp (153% of the Democrat excess death rate). The gap in
excess death rates between Republicans and Democrats is concentrated in
counties with low vaccination rates and only materializes after vaccines
became widely available.

Coronavirus disease 2019 (COVID-19) has caused over one million deaths in the
United States [6], leading to considerable interest in identifying risk
factors for COVID-19 mortality. Political affiliation has emerged as one
potentially significant risk factor, amid evidence that Republican-leaning
counties have had higher COVID-19 death rates than Democrat-leaning counties
[11, 14]. But it is unknown whether this county-level association — which may
be subject to the ecological fallacy [12] — persists at the individual level.
Prior research has also established differences in vaccination attitudes and
social distancing based on political party affiliation [1, 9, 2, 8, 13], but
it has been more difficult thus far to establish corresponding links to health
outcomes due to data limitations. This study overcomes that challenge by
linking voter registration data in Ohio and Florida to mortality data to
assess the individual-level association between political party affiliation
and excess mortality during the COVID-19 pandemic. We estimate higher excess
death rates for registered Republicans when compared to registered Democrats
after vaccines were widely available — and not before — and these differences
were concentrated in counties with lower vaccination rates.

## Results

To calculate excess deaths, we use 577,659 deaths of individuals linked to
their 2017 voting records in Ohio and Florida who died at age 25 or older
between January 2018 and December 2021. Our approach estimates “excess death
rates” as the percent increase in deaths above expected deaths that are due to
seasonality, geographic location, party affiliation, and age. These expected
deaths are calculated non-parametrically using 2019 data by aggregating deaths
into counts $N_{mcpa,2019}$ at the month-by-county-by-party-by-age-bin level.
The age bins used were 25-64, 65-74, 75-84, and 85-and-older. We then estimate
relative excess deaths $E_{mcpa,t}$ in other periods $t$ for a given county-
by-party-by-age cell by considering the percent change in deaths:

$E_{mcpa,t}=(N_{mcpa,t}-N_{mcpa,2019})\big{/}N_{mcpa,2019}.$ (1)

We also report aggregated excess death measures (e.g., the percent increase in
excess deaths in 2020 and 2021) by taking the average of $E_{mcpa,t}$,
weighted by $N_{mcpa,2019}$. Thiscontrols for differences in _pre_ -COVID-19
death counts across calendar month, county of residence at time of voter
registration, political party registration (Democrat or Republican), and age
bins.

Figure 1: The figure plots monthly excess deaths for Florida and Ohio based on
mortality records linked to voter registration files. The first panel plots
aggregate excess death rates. The second panel plots excess death rates
separately for registered Republicans and Democrats. The third panel plots the
percentage point difference in excess deaths between Republicans and Democrats
after regression adjusting for year-month-by-age-bin-by-county differences.
The error bars reflect the pointwise 95% confidence interval.

In Figure 1, we plot the values of $E_{mcpa,t}$ by months for 2018, 2020, and
2021 in different ways. In the first panel we report the percent excess deaths
in aggregate, pooling registered Democrats and Republicans. The aggregate
excess death rate from March 2020 to December 2021 was 10.3 percentage points
(pp) (95% CI:6.7,14.0) (1). Calendar year 2018 and the beginning of 2020 serve
as a placebo test for our method, as excess death rates should be
approximately zero prior to the COVID-19 pandemic. As expected, Figure 1
depicts only small fluctuations in excess death rates around zero prior to the
pandemic. Finally, the qualitative patterns of excess deaths we observe in
Florida and Ohio during this time match the trends reported elsewhere, but the
magnitudes are slightly smaller [5].

In the second panel of Figure 1, we plot excess death rates for Republicans
and Democrats separately. In 2018 and the early parts of 2020, excess death
rates for Republicans and Democrats are similar, and centered around zero.
Both groups experienced a similar large spike in excess deaths in the winter
of 2020-2021. However, in the summer of 2021 — after vaccines were widely
available — the Republican excess death rate rose to nearly double that of
Democrats, and this gap widened further in the winter of 2021.

In the third panel of Figure 1, we estimate the difference between Republican
and Democrat excess death rates by regressing $E_{mcpa,t}$ on Republican-year-
month indicators, controlling for year-month-by-county-by-age-bin fixed
effects. These controls adjust for differences in excess death rates _during_
COVID-19, in addition to the adjustments for death rates by age, party and
location from the construct of excess death. We report the point estimates and
95% confidence intervals for each of these estimates (standard errors are
clustered by county). A joint test of the null hypothesis that the
coefficients in 2018 and January and February of 2020 are equal to zero fails
to reject ($p$ = 0.62). Between March 2020 and December 2021, the part of our
study period that overlaps the COVID-19 pandemic, average excess death rates
were 5.4 pp (76%) higher among Republicans than Democrats (95% CI:2.9,8.0;
$p\leq 0.001$). However, this gap widened from 1.6 pp (95% CI:-1.2,4.4;
$p=0.25$) between March 2020 and March 2021, to a 10.6 pp (95% CI:7.3,13.8;
$p\leq 0.001$), or 153%, difference after April 5, 2021, when all adults were
eligible for COVID-19 vaccines in Florida and Ohio.

This sharp contrast in the excess death rate gap before and after vaccines
were available suggests that vaccine take-up likely played an important role.
Data on vaccine take-up by party is limited and unavailable in our dataset,
but there is evidence of differences in vaccination attitudes and reported
uptake based on political party affiliation [13, 10, 7]. Using county-level
vaccination rates, we find evidence that vaccination contributes to explaining
differences in excess deaths by political party affiliation, even after
controlling for location and age differences.

Figure 2: The figure plots the association between a one standard deviation
increase in the county-level vaccination rate (roughly 10 percent) on the
Republican-Democrat monthly excess death rate gap in Florida and Ohio. The
coefficients regression adjust for year-month-by-age-bin-by-county
differences. The error bars report the 95% confidence interval.

We reexamine the Republican-Democrat excess death rate gap from Figure 1, but
now interact Republican-year-month indicators with a measure of county-level
vaccination rates (the number of individuals with at least one dose scaled by
population) as of June 1, 2021. This measure is scaled to have a standard
deviation (SD) of one, so the coefficient is the association between a one SD
increase in the county-level vaccination rate (roughly 10 percent) and the
magnitude of the Republican-Democrat difference in excess death rates. In
Figure 2, we report these monthly coefficients over time, along with 95%
confidence intervals. We do not observe a statistically significant
association between the county-level vaccination rate and the Republican-
Democrat excess death gap until after the vaccine is widely available, with a
sharp (and increasingly) negative effect toward the end of sample. This
suggests that during the period after vaccines were widely available, areas
with larger take-up of vaccines saw much smaller excess death rate gaps
between Republicans and Democrats (even after adjusting for county and age
differences).

Figure 3: The figure plots average excess death rates for Republicans and
Democrats in each county for Covid Pre-Vaccine (April 2020 to March 2021) and
Covid Post-Vaccine (April 2021-December 2021). The circles are each county,
and diamonds are binned means [3] for each party. The curves are quadratic
fits using least squares.

In Figure 3, we explore the association between county-level vaccination rates
and excess death rates for Republicans and Democrats before and after vaccines
were widely available. We pool our excess death data into two time periods:
Covid “Pre-Vaccine” (April 2020 to March 2021) and Covid “Post-Vaccine” (April
2021-December 2021), referring to the period when vaccines were available to
all adults in Florida and Ohio. For each time period, we calculate the average
excess death rates for Republicans and Democrats in each county, and then
calculate binned means of excess death rate [3] for each party. Finally, we
fit quadratic curves for each group using least squares.

Two noteworthy facts emerge in Figure 3. First, in the Covid Pre-Vaccine
period, the association between excess death rates and county-level
vaccination rates are nearly identical for Democrats and Republicans.
Statistically, we fail to reject differences in both slopes and intercept
between the two groups in this period ($p$ = 0.73 for slopes and $p$ = 0.84
for slopes and intercepts). Second, in the Covid Post-Vaccine period, there is
a clear visual difference between Democrats and Republicans ($p$ =0.02 for
slopes and $p$ = 0.00 for slopes and intercepts), with higher excess death
rates for Republicans in counties with lower vaccination rates. By comparison,
the difference in excess deaths between Republicans and Democrats is nearly
zero in counties with the highest vaccination rates.

## Discussion

Political party affiliation was associated with excess death rates at the
individual level during the initial years of the COVID-19 pandemic. Registered
Republicans in Florida and Ohio had higher excess death rates than registered
Democrats, driven by a large mortality gap in the period after all adults were
eligible for vaccines. These results adjust for county-by-age differences in
excess deaths during the pandemic, suggesting that there were within-age-by-
county differences in excess death associated with political party
affiliation.

Our study has several limitations. First, our mortality data, while detailed
and recent, only includes approximately 80% of deaths in the US. However,
excess death patterns in our data are similar to those in other reliable
sources. Second, because we did not have information on an individual’s
vaccination status, analyses of the association between vaccination rates and
excess deaths relied on county-level vaccination rates. Third, our study is
based on data from the only states where we could obtain voter registration
information (Florida and Ohio); hence, our results may not generalize to other
states.

Overall, our results suggest that political party affiliation only became a
substantial risk factor in Ohio and Florida after vaccines were widely
available. Lack of individual-level vaccination status limits our ability to
draw broad conclusions, but the results suggest that the well-documented
differences in vaccination attitudes and reported uptake between Republicans
and Democrats [10, 7, 8, 13] have already had serious consequences for the
severity and trajectory of the pandemic in the United States. If these
differences in vaccination by political party affiliation persist, then the
higher excess death rate among Republicans is likely to continue through the
subsequent stages of the COVID-19 pandemic.

## Materials and Methods

#### Data

Detailed mortality data for 2018-2021 was obtained from Datavant, an
organization that uses privacy-preserving record linkage to connect the Social
Security Administration Death Master File with information from newspapers,
funeral homes, and other sources to construct an individual-level database
with over 80% of annual US deaths. This data was linked at the individual
level to 2017 Florida and Ohio voter registration files [15]. For each record,
the linked data include month of death, age of deceased, county of residence,
and 2017 political party registration. Political party affiliation in Florida
is actively registered by voters. Political party affiliation in Ohio is
defined by whether they voted in a party’s primary election within the
preceding two calendar years. We excluded individuals that were registered
independents or affiliated with third parties. We obtained county-level
vaccination rates as of June 1, 2021 from the Centers for Disease Control and
Prevention [4].

A web appendix, aggregated data, and analysis scripts will be deposited in the
Open Science Framework (https://osf.io/m42d7/). Researchers interested in
using individual-level data will need to acquire the mortality data from
Datavant. The study was deemed exempt by the Yale University IRB.

#### Statistical measurement

To construct an estimate of excess death rates, we aggregated death counts at
the month-by-county-by-party-by-age level. The age bins used were 25-64,
65-74, 75-84, and 85+. We calculated percent increases in deaths by calendar
month in 2018, 2020 and 2021, relative to 2019, for each county-by-party-by-
age cell. When plotting aggregated excess death rates by month, we take
averages weighted by baseline death counts. We then estimated differences in
excess death rates by period and political party affiliation, plotted them,
and assessed their association with county-level vaccination rates.
Statistical models adjusted for differences in excess death rates by calendar
quarter, county, and age.

Measurement error does not appear to affect our conclusions of the excess
death rate gap. First, in Figure 1 the gap in rates is statistically
indistinguishable from zero in 2018 and the beginning of 2020, suggesting no
correlation between any measurement error and party affiliation. Second, to
the extent we undermeasure deaths during the COVID-19 period and it is
uncorrelated with party affiliation, then we still correctly estimate the
relative _percentage_ gap of 76 percent.

## References

  * [1] H. Allcott, L. Boxell, J. Conway, M. Gentzkow, M. Thaler, and D. Yang.  Polarization and public health: Partisan differences in social distancing during the coronavirus pandemic.  Journal of public economics, 191:104254, 2020.
  * [2] T. Callaghan, A. Moghtaderi, J. A. Lueck, P. Hotez, U. Strych, A. Dor, E. F. Fowler, and M. Motta.  Correlates and disparities of intention to vaccinate against covid-19.  Social science & medicine (1982), 272:113638, 2021.
  * [3] M. D. Cattaneo, R. K. Crump, M. H. Farrell, and Y. Feng.  On binscatter.  arXiv preprint arXiv:1902.09608, 2019.
  * [4] Covid-19 vaccinations in the united states,county.  https://data.cdc.gov/Vaccinations/COVID-19-Vaccinations-in-the-United-States-County/8xkx-amqh, 2022\.  Accessed: 2022-06-08.
  * [5] Excess deaths associated with covid-19.  https://www.cdc.gov/nchs/nvss/vsrr/covid19/excess_deaths.htm, Sep 2022.  Accessed: 2010-09-15.
  * [6] National center for health statistics. provisional death counts for coronavirus disease 2019 (covid-19). covid-19 mortality overview, May 2022.
  * [7] Coronavirus: Vaccinations in civiqs.  https://civiqs.com/results/coronavirus_vaccine, 9 2020.  Accessed: 2020-09-20.
  * [8] S. K. Cowan, N. Mark, and J. A. Reich.  Covid-19 vaccine hesitancy is the new terrain for political division among americans.  Socius, 7:23780231211023657, 2021.
  * [9] G. Grossman, S. Kim, J. M. Rexer, and H. Thirumurthy.  Political partisanship influences behavioral responses to governors’ recommendations for covid-19 prevention in the united states.  Proceedings of the National Academy of Sciences, 117(39):24144–24153, 2020.
  * [10] A. Kirzinger.  Kff covid-19 vaccine monitor: The increasing importance of partisanship in predicting covid-19 vaccination status.  Technical report, Kaiser Family Foundation, Nov 2021.
  * [11] D. Leonhardt.  The partisan gap in u.s. covid deaths is still growing, but more slowly.  The New York Times, 2 2022.
  * [12] S. Piantadosi, D. P. Byar, and S. B. Green.  The ecological fallacy.  American journal of epidemiology, 127(5):893–904, 1988.
  * [13] S. L. Pink, J. Chu, J. N. Druckman, D. G. Rand, and R. Willer.  Elite party cues increase vaccination intentions among republicans.  Proceedings of the National Academy of Sciences, 118(32):e2106559118, 2021.
  * [14] N. J. Sehgal, D. Yue, E. Pope, R. H. Wang, and D. H. Roby.  The association between covid-19 mortality and the county-level partisan divide in the united states.  Health Affairs, 41(6):853–863, 2022.
  * [15] G. Sood.  Florida Voter Registration Data (2017 and 2022), 2017.

Supplementary Information for

“Excess death rates for Republicans and Democrats during the COVID-19
pandemic”

##  Appendix A Supplemental description of methods and data

###  A.1 Description of study data and linkages

This study made use of data from four different sources. Below, we describe
each of these data sources in more detail and then discuss how these data were
assembled into our analytic file. Portions of the dataset will be publicly
available on an OSF repository upon publication: https://osf.io/m42d7/ These
include replication code using aggregated data, aggregated data for our two
main datasets used in the paper, the Ohio Voter File, and this supplement.

#### Florida voter file:

The publicly available Florida voter file for February 2017 was accessed via
the Harvard Dataverse. For additional details on the file and a link to
request access to the data proceed to the following:
https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/UBIG3F.
The file contains full name, date of birth, county of registration, gender,
and party affiliation.

#### Ohio voter file:

The publicly available Ohio voter file for 2017 was accessed via the Ohio
Secretary of State website at:
https://www6.ohiosos.gov/ords/f?p=VOTERFTP:HOME. We accessed the link and
downloaded the data on March 4, 2017. The file contains full name, date of
birth, county of registration, and party affiliation.

#### Datavant

: Detailed mortality data for 2018 to 2021 was obtained from Datavant, an
organization that uses privacy-preserving record linkage to connect the Social
Security Administration Death Master File with information from newspapers,
funeral homes, and other sources to construct an individual-level database
with over 80% of annual US deaths. For each record, the identifiable data
indicate the month of death and age of the deceased individual in months, as
well as individual-level identifiers that could be used to link to the voter
files but that we, as researchers, did not have access to in the linked data.
The Datavant mortality data was then linked, at the individual level, to the
Florida and Ohio voter files on first name, last name, and date of birth using
a proprietary algorithm. A match indicated that a registered voter in Florida
and Ohio had been identified in the Datavant mortality data and provided
detailed information on date of death and the age of the deceased individual
in months, top-coded at 89 years old. A deidentified version of the linked
data was then provided to our research team. That dataset contained date of
death, year-month of birth, county of residence based on the voter
registration file in 2017, party affiliation, and either gender (if from
Florida) or a probabilistic guess at gender (if from Ohio) based on first
name.

#### Center for Disease Control and Prevention (CDC):

We obtained additional information from the CDC. From the CDC, we also
accessed data on county-level vaccination rates as of June 23, 2022. That data
can be accessed here: https://data.cdc.gov/Vaccinations/COVID-19-Vaccinations-
in-the-United-States-County/8xkx-amqh. From that data, we obtained information
on what share of a county’s population had received at least one dose of a
Covid-19 vaccine as of June 6, 2021. We selected this date—two months after
vaccines became available to all adults in our study states—because it
represented the approximate time when all adults had had the opportunity (if
they so desired) to complete the two-dose vaccine series as well as the
additional 14 days following the second dose needed to be considered “fully
vaccinated” according to the CDC.

###  A.2 Methodology for calculating excess death rates

To construct an estimate of excess death rates, we first aggregated death
counts at the month-by-county-by-party-by-age level. The age bins that we used
were: 25-64, 65-74, 75-84, 85+. We kept registered Republicans and Democrats
and removed all unaffiliated or independent voters, as well as voters
registered to any other political parties.

For each month in 2018, 2020, and 2021, we then calculated the percent
increase in deaths relative to 2019. While our aggregated data is at the
month-by-county-by-party-by-age level, we map months to quarters (e.g.,
January-March to quarter 1) to coarsen our cells and reduce noise in
estimating the expected monthly death count from which excess deaths in 2018,
2020, and 2021 are constructed. Effectively, this means that all the months in
a quarter have the same predicted death count conditional on a particular
county-by-party-by-age cell. For example, we would measure the excess death
rate for 25-64 year old Republicans registered in Franklin county (in Ohio) in
each month of 2020 by dividing the count of deaths for individuals in that
group by the average monthly death count in the first quarter of 2019 for that
age cell (25-64 year old Republicans in Franklin county in Ohio). Intuitively,
this measure of excess deaths adjusts for baseline (i.e., pre-COVID-19)
differences in mortality by county, registered political party, calendar
quarter, and age bins. When we plot the aggregated data for a particular month
(Figure), we take a weighted average of the excess death rates in each of the
cells, weighted by the 2019 death count.

###  A.3 Statistical model

To assess overall excess death rates during the COVID-19 pandemic, and to
adjust differences in excess death rates by political party affiliation for
age bins and county of residence, we used multivariable linear regression
models. In some cases, we ran these regressions separately by time period. We
selected April 5, 2021 as a cutoff for when vaccines were widely available to
adults in Florida and Ohio. The federal deadline for when states had to open
eligibility for vaccines for all adults was April 24, 2021, but Florida and
Ohio did so earlier. Florida made vaccines available to all adults on April 5,
2021. For more details on Florida’s policy see:
https://floridahealthcovid19.gov/latest-vaccine-updates/. Ohio made vaccines
available to all adults earlier, on March 29, 2021. For more details on Ohio’s
policy see: https://governor.ohio.gov/media/news-and-media/expanded-vaccine-
eligibility-cleveland-mass-vaccination-clinic-opens-to-public-03162021. Hence,
we selected the latter of the two dates, April 5, 2021, as the date by which
all adults in both states were eligible for vaccines. Because our data is
monthly, we use April 2021 data on and onwards to reflect this period, which
corresponds closely to when vaccines were widely available.

To estimate the difference in excess death rates by month (the third panel of
Figure 1), regression models were of the form:

$Y_{tcpa}=\gamma_{tca}\times W_{tca}+\beta_{t}\times 1(\text{Affiliation =
Republican})_{p}\times 1(\text{Year-month}=t)_{t}+\epsilon_{tcpa}$

where $W_{tca}$ is a set of fixed effects for year-month-by-county-by-age-bin,
and we plot the $\beta_{t}$ coefficients over time (marking the relative
difference in excess deaths for Republicans).

We also estimated regression models that pooled data from multiple months and
recovered a single coefficient for three different periods of the COVID-19
pandemic: April 1, 2020 – December 31, 2021; April 1, 2020 – March 31, 2021;
and April 1, 2021 – December 31, 2021. When estimating overall excess death
rates, these regressions were simply estimates of the constant in a
regression:

$Y_{tcpa}=\beta_{0}+\epsilon_{tcpa}$

where the coefficient of interest is $\beta_{0}$.

To estimate the effect of vaccination rates on the difference in excess death
rates by month (the third panel of Figure 1), regression models were of the
form:

$\displaystyle Y_{tcpa}$ $\displaystyle=\gamma_{tca}\times
W_{tca}+\beta_{t}\times 1(\text{Affiliation = Republican})_{p}\times
1(\text{Year-month}=t)_{t}$ $\displaystyle+\tau_{t}\times 1(\text{Affiliation
= Republican})_{p}\times 1(\text{Year-
month}=t)_{t}\times\text{CountyVaccionationRate}_{c}+\epsilon_{tcpa}$

where $W_{tca}$ is a set of fixed effects for year-month-by-county-by-age-bin,
$CountyVaccineRate_{c}$ is the share of the county’s population that had at
least one dose of vaccines as of June 2021, and we plot the $\tau_{t}$
coefficients over time (marking the change in the relative difference in
excess deaths for Republicans when increasing county vaccination by one
standard deviation ( 10 percent)).

In all regression cases, we cluster standard errors by county.

In the last Figure, we calculate the average excess death rates by county for
Democrats and Republicans between April 2020 to March 2021 and April 2021 to
December 2021. We plot the county level measures and fit a quadratic curve for
each group and time period. We also plot binned means using the binsreg
package provided by Cattaneo et al. (2021).