# Using knot Floer invariants to detect prime knots

Samantha Allen  Department of Mathematics and Computer Science, Duquesne
University, Pittsburgh, PA 15282<EMAIL_ADDRESS>https://samantha-
allen.github.io/ ,  Charles Livingston  Department of Mathematics, Indiana
University, Bloomington, IN 47405<EMAIL_ADDRESS>https://livingst.pages.iu.edu/ ,  Misha Temkin  Department of Mathematics,
Dartmouth College, Hanover, NH 03755<EMAIL_ADDRESS>https://math.dartmouth.edu/~mt/ and  C.-M. Michael Wong  Department of
Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5
<EMAIL_ADDRESS>https://mysite.science.uottawa.ca/cwong/

###### Abstract.

We present knot primality tests that are built from knot Floer homology. The
most basic of these is a simply stated and elementary consequence of Heegaard
Floer theory: if the knot Floer polynomial $\Omega_{K}(s,t)$ is irreducible,
then $K$ is prime. Improvements in this test yield a primality condition that
has been over 90% effective in identifying prime knots of up to 30 crossings.
As another illustration of the strength of these tools, there are 1,315 non-
hyperbolic prime knots with crossing number 20 or less; the tests we develop
prove the primality of over 96% of them.

The filtered chain homotopy class of $\widehat{\operatorname{CFK}}(K)$ has a
unique minimal-dimension representative that is the direct sum of a one-
dimensional complex and two-dimensional complexes, each of which can be
assigned a parity. Let $\delta(K)$, $b_{\mathrm{e}}(K)$, and
$b_{\mathrm{o}}(K)$ denote the dimension of this minimal representative and
the number of even and odd two-dimensional summands, respectively. For a
composite knot $K$, we observe that there is a non-trivial factorization
$\delta(K)=d_{1}d_{2}$ satisfying

$(d_{1}-1)(d_{2}-1)\leqslant 4\min(b_{\mathrm{e}}(K),b_{\mathrm{o}}(K)).$

This yields another knot primality test. One corollary is a simple proof of
Krcatovich’s result that $L$-space knots are prime.

##  1\. Introduction

The main tools for proving that a knot $K\subset S^{3}$ is prime have been
geometric, depending on hyperbolic geometry, incompressible surfaces, or
geometric decompositions of the knot complement. Background is provided in
[6]; more recent discussions include [7, 28, 29]. Here we will explore
algebraic knot primality tests based on Floer knot theory. For example, the
most basic of these tests is a quick consequence of Heegaard Floer theory: If
the two-variable knot Floer polynomial $\Omega_{K}(s,t)$ built from
$\widehat{\operatorname{HFK}}(K)$ is irreducible, then $K$ is prime.

In the case that one has an alternating diagram for $K$, Menasco [17] gave
simple necessary and sufficient conditions that determine if $K$ is prime.
More relevant to our work is the fact that if a knot is alternating, then the
degree of its Alexander polynomial is twice its genus [8, 19, 20]. As a
consequence, there is a primality test for alternating knots $K$ in terms of
the Alexander polynomial: If $K$ is alternating and $\Delta_{K}(t)$ is
irreducible, then $K$ is prime. The primality tests we describe generalize
this to all knots and are much stronger, even in the case of alternating
knots.

We will be working with the knot Floer chain complex
$\widehat{\operatorname{CFK}}(K)$ and its associated graded homology
$\widehat{\operatorname{HFK}}(K)$, first defined in [23]. We will work
exclusively with coefficients in the field with two elements,
$\mathbb{F}_{2}$. The computation of this complex is algorithmic [25, 26]. A
program written by Zoltán Szabó, enhanced by Jonathan Hanselman, and now
installed in SnapPy [9] has been used to develop examples.

In the next section we present primality tests that are built using
$\widehat{\operatorname{HFK}}(K)$. A complete survey of prime knots with
crossing number at most 15 and a large random sample of knots through 30
crossings indicate that these tests prove primality for over 90% of prime
knots. Surprisingly, the tests appear to be slightly stronger for non-
alternating knots. In Section 3 we consider the filtered chain complex
$\widehat{\operatorname{CFK}}(K)$, taking advantage of its differential as
well as its homology. The tests using this added structure are more
specialized; so far they appear to be of theoretical, rather than
computational, interest. For instance, this approach offers a simple proof of
Krcatovich’s theorem that $L$–space knots are prime [12]; see Corollary 7.

### Acknowledgments

The authors thank Zoltán Szabó and Jonanthan Hanselman for their pointers
regarding the use of Zoltán’s knot Floer program. The authors appreciate
Nathan Dunfield’s guidance regarding SnapPy. Feedback from Peter Ozsváth led
us to further explore the case of alternating knots.

SA thanks the Wimmer Family Foundation for their support. CMMW acknowledges
the support of the Natural Sciences and Engineering Research Council of Canada
(NSERC), RGPIN-2023-05123.

##  2\. Knot Floer homology primality obstructions

Let $\widehat{\operatorname{CFK}}(K)$ denote the knot Floer homology complex
[23] with coefficients in the field with two elements. This is a finite-
dimensional, graded, filtered chain complex that is well-defined up to
filtered chain homotopy equivalence. We let
$\widehat{\operatorname{CFK}}(K,i)$ denote the subcomplex at filtration level
$i\in\mathbb{Z}$. The group $\widehat{\operatorname{HFK}}_{j}(K,i)$ is the
homology of the quotient complex
$\widehat{\operatorname{CFK}}(K,i)/\widehat{\operatorname{CFK}}(K,i-1)$ at
grading $j$. We define the knot Floer polynomial by

$\Omega_{K}(s,t)=\sum\dim(\widehat{\operatorname{HFK}}_{j}(K,i))s^{j}t^{i}.$

A feature of knot Floer homology is that it is multiplicative:
$\widehat{\operatorname{CFK}}(K\mathbin{\sharp}J)=\widehat{\operatorname{CFK}}(K)\otimes\widehat{\operatorname{CFK}}(J)$.
From this it follows that
$\Omega_{K\mathbin{\sharp}J}(s,t)=\Omega_{K}(s,t)\Omega_{J}(s,t)$. (See [16]
for a brief discussion of multiplicativity; we will present a few details in
the appendix.) Futhermore, Ozsváth and Szabó proved in [22] that
$\widehat{\operatorname{HFK}}(K)$ detects the genus of a knot; in particular,
$\Omega_{K}(s,t)=1$ if and only if $K$ is the unknot. Thus we have a basic
primality test.

######  Proposition 1.

If $\Omega_{K}(s,t)$ is irreducible, then $K$ is prime.

Note there are algorithms for determining whether or not a multivariable
polynomial over the integers is prime.

This result provides a primality test that has been effective in roughly 78%
of our experiments, which have included prime knots with up to 30 crossings.
There are a few refinements that make it more effective, as we now describe.

According to [23], the polynomial $\Omega_{K}(s,t)=\sum c_{i,j}s^{j}t^{i}$
satisfies the symmetry condition $c_{i,j}=c_{-i,j-2i}$ for all $i$ and $j$. We
say that a factorization of a two-variable Laurent polynomial is symmetric if
each factor satisfies this symmetry condition. If no such nontrivial
factorization exists, the polynomial is called symmetrically irreducible.

######  Theorem 2.

If $\Omega_{K}(s,t)$ is symmetrically irreducible, then $K$ is prime.

It is known that knot Floer homology detects the torus knots $T(2,3)$ and
$T(2,5)$, as well as the figure eight knot, $4_{1}$ (see [10, 11]). Here is a
list of all nontrivial knots that are known to be determined by
$\widehat{\operatorname{CFK}}(K)$, along with their knot Floer polynomials
$\Omega_{K}(s,t)$.

  * •

$\ \ T(2,3)\colon\thinspace\ \ \ t+s^{-1}+s^{-2}t^{-1}$.

  * •

$-{T(2,3)}\colon\thinspace\ \ s^{2}t+s+t^{-1}$.

  * •

${\ T(2,5})\colon\thinspace\ \ \
t^{2}+s^{-1}t+s^{-2}+s^{-3}t^{-1}+s^{-4}t^{-2}$.

  * •

$-{T(2,5})\colon\thinspace\ \ s^{4}t^{2}+s^{3}t+s^{2}+s^{1}t^{-1}+t^{-2}$.

  * •

${\ 4_{1}}\colon\thinspace\ \ \ \ \ \ \ \ \ st+3+s^{-1}t^{-1}$.

######  Theorem 3.

If $K$ is composite and $\Omega_{K}(s,t)$ factors into exactly two
symmetrically irreducible factors, one of which is in the list above, then $K$
has a factor that is the corresponding knot in the list.

In practice, strong tools are available that often can be used to prove that a
knot $K$ does not have any of these five knots as factors; multiplicative
invariants such as the Jones polynomial are sufficient, though among prime
knots with at most 15 crossings, we did find a few examples in which the
HOMFLY polynomial (see [13]) was required.

Summary of computations.

In a complete survey of prime knots of 15 or fewer crossings (of which there
are 313,230), Theorem 2 proved the primality of just under $80\%$ of them.
Applying Theorem 3 increased that percentage to over 90%. Random sampling of
prime knots with 25 to 30 crossings showed that a success rate of above 90%
persists to higher crossings. It appears that the tests are slightly more
effective for non-alternating knots, rising to be above 92% successful. Since
non-alternating knots dominate for higher crossing numbers, we expect that our
tests will generally be over 92% effective.

In the enumeration of prime knots, which has now been carried out through 20
crossings independently by Burton and Thistlethwaite, one of the most
challenging aspects has been in proving the primality of those that are not
hyperbolic. Papers on this topic include [7, 29, 28]. In that enumeration, of
the more than two billion such knots, there are 1,315 that are not hyperbolic,
including nine torus knots. Applying Theorem 3 proves the primality of 1,269
of them, yielding a success rate of over 96%.

##  3\. Bar-complexes and further primality obstructions

An elementary homological result states that any finite-dimensional, graded,
filtered chain complex is filtered chain homotopy equivalent to an essentially
unique bar-complex. Here, by bar-complex we mean a complex along with a direct
sum decomposition into one-dimensional complexes $(F_{i})$ and two-dimensional
complexes $(T_{i}\to B_{i})$ which are acyclic as chain complexes and for
which the filtration level of $B_{i}$ is strictly less than that of $T_{i}$.
In the language that is now common in persistent homology literature,
intervals bounded by the filtration levels of $B_{i}$ and $T_{i}$ are called
bars. To simplify the discussion, we will modify this notation and call the
two-dimensional summands bars. A bar is called even or odd depending on
whether the grading of $B_{i}$ is even or odd, respectively.

Proofs of the existence and uniqueness of bar-complexes appear in such sources
as [4, 27]; going further back, see also [18]. In forthcoming work, the
authors will present a more general use of persistent homology to study the
doubly filtered complex $\widehat{\operatorname{CFK}}(K)$; see [2] for
background. In knot Floer theory literature, the bar-complex appears in the
context of a reduced complex with a vertically simplified basis; see, for
example, [14, Definition 11.23]. Here is a brief summary, in spirit close to
that in [4].

Choose an element $x$ of least filtration level for which $\partial x=y\neq
0$. One can choose a filtered basis so that $x$ and $y$ are elements of that
basis. By adding $x$ to other filtered basis elements we can ensure that $x$
is the only basis element whose boundary has a nontrivial $y$–coefficient. The
fact that $\partial^{2}=0$ implies that the pair $\\{x,y\\}$ generates a
summand. We now have that $x$ and $y$ split off a two-dimensional summand. If
the filtration levels are the same, removing the summand is a filtered chain
homotopy equivalence. If the filtration levels are different, we have split
off a bar.

The number of bars which pair elements of specified filtrations levels can be
expressed in terms of homological information derived from the maps
$H_{*}(\widehat{\operatorname{CFK}}(K,i))\to
H_{*}(\widehat{\operatorname{CFK}}(K,j))$. This implies the uniqueness of the
bar-complex.

###  3.1. Bar-complexes for $\widehat{\operatorname{CFK}}(K)$ and connected
sums

Tensor products of bar-complexes arise naturally when considering connected
sums of knots. Thus, we will be taking advantage of the fact that the tensor
product of two bar-complexes is a bar-complex. More precisely, we have the
following lemma, the proof of which is an easy computation that we leave to
the reader.

######  Lemma 4.

The tensor product of bars, $(T_{1}\to B_{1})\otimes(T_{2}\to B_{2})$, is
isomorphic to the direct sum of bars, $(X_{1}\to Y_{1})\oplus(X_{2}\to
Y_{2})$, where the grading and filtration levels of $Y_{1}$ are the sums of
those for $B_{1}$ and $B_{2}$ and the grading and filtration levels of $X_{2}$
are the sums of those for $T_{1}$ and $T_{2}$. In particular, one of the
summands is even and one is odd.

As an unfiltered complex, the homology of $\widehat{\operatorname{CFK}}(K)$ is
one-dimensional, supported in grading 0. (The corresponding filtration level
is the $\tau$–invariant defined by Ozsváth–Szabó in [21].) For a knot $K$, let
$\widehat{C}(K)$ denote the bar-complex representative of
$\widehat{\operatorname{CFK}}(K)$. We have the following definitions.

  * •

$\delta(K)=\dim\widehat{C}(K)$. One can see that $\delta(K)$ is the total
dimension of $\widehat{\operatorname{HFK}}(K)$.

  * •

$b_{e}(K)$ and $b_{o}(K)$ are the number of even and odd bars, respectively,
in $\widehat{C}(K)$. Note that $\delta(K)=1+2\left(b_{e}(K)+b_{o}(K)\right)$.

An easy counting argument proves the following result.

######  Theorem 5.

For any two knots $K_{1}$ and $K_{2}$,
$\delta(K_{1}\mathbin{\sharp}K_{2})=\delta(K_{1})\delta(K_{2})$. Furthermore,

$\displaystyle b_{\mathrm{e}}(K_{1}\mathbin{\sharp}K_{2})$
$\displaystyle=b_{\mathrm{e}}(K_{1})+b_{\mathrm{e}}(K_{2})+\frac{1}{4}(\delta(K_{1})-1)(\delta(K_{2})-1);$
$\displaystyle b_{\mathrm{o}}(K_{1}\mathbin{\sharp}K_{2})$
$\displaystyle=b_{\mathrm{o}}(K_{1})+b_{\mathrm{o}}(K_{2})+\frac{1}{4}(\delta(K_{1})-1)(\delta(K_{2})-1).$

Here is an evident corollary.

######  Corollary 6.

If $K$ is composite, then there is a nontrivial factorization
$\delta(K)=d_{1}d_{2}$ for which

$(d_{1}-1)(d_{2}-1)\leqslant 4\min(b_{\mathrm{e}}(K),b_{\mathrm{o}}(K)).$

###  3.2. Applications of Corollary 6

$L$–space knots are prime. An immediate consequence is a theorem of Krcatovich
[12, Theorem 1.2]; subsequent proofs have appeared in [3, 5].

######  Corollary 7 (Krcatovich).

$L$-space knots are prime.

###### Proof.

It follows from the proof of [24, Theorem 1.2] that if $K$ is an $L$-space
knot, then $b_{o}(K)=0$. Here are a few details. The theorem states that there
is a filtered basis for $\widehat{\operatorname{CFK}}(K)$ consisting of single
generators at a strictly increasing set of filtration levels
$f_{0},f_{1},\ldots,f_{2k}$, for some $k$. The gradings of these generators
alternate in parity, are strictly increasing, and the top level generator is
at grading 0. The concluding paragraph of the proof of [24, Theorem 1.2] shows
that the homology group $H(\widehat{\operatorname{CFK}}(K,f_{i}))=0$ for $i$
odd. Thus, the bar-complex consists of a single generator summand at
filtration level $f_{2k}$ and bars $(T_{i}\to B_{i})$ where $T_{i}$ is at
filtration level $f_{2i+1}$ and $B_{i}$ is at filtration level $f_{2i}$ for
$i=0,...,k-1$. Each of these bars is even. ∎

######  Remark 8.

Jonathan Hanselman has informed the authors that a converse holds: If
$b_{\mathrm{o}}(K)=0$, then $K$ is an $L$-space knot.

Successful applications of Corollary 6 to low-crossing prime knots are rare.
In fact, the only prime knots of 12 or fewer crossings that have primality
detected by Corollary 6 are $L$–space knots. For other low-crossing knots, the
values of $b_{\mathrm{e}}(K)$ and $b_{\mathrm{o}}(K)$ are close in value, both
roughly $\delta(K)/4$; Corollary 6 can be applied when one of the two is close
to 0.

From our experiments, it seems that $b_{\mathrm{e}}(K)-b_{\mathrm{o}}(K)$ is
related to the dealternating number of $K$; that is, the minimum number of
crossing changes required to convert $K$ into an alternating knot. (References
for the dealternating number include [1, 15, 30].) These examples provide
little evidence upon which to base a conjecture and the topic remains open for
future study.

##  Appendix A Multiplicativity of $\Omega_{K}(s,t)$.

The existence of a bar-complex representative of
$\widehat{\operatorname{CFK}}(K)$ implies the simpler fact that
$\widehat{\operatorname{CFK}}(K)$ has a reduced representative, one for which
the boundary map strictly lowers filtration levels. Let $\widehat{C}(K)$
denote such a reduced complex with an ordered, graded, filtered basis. Then
the coefficient $c_{i,j}$ of $s^{j}t^{i}$ in $\Omega_{K}(s,t)$ is precisely
the number of basis elements at grading $j$ and at filtration level $i$.

Let $\widehat{C}^{1}$ and $\widehat{C}^{2}$ be two such based reduced knot
complexes. Their associated polynomials have coefficients which we denote by
$c_{i,j}^{\epsilon}$ for $\epsilon=1,2$. The tensor product of reduced
complexes is clearly reduced. Denote the tensor product by $\widehat{D}$ and
denote the coefficients of its associated polynomial by $d_{i,j}$. A simple
counting argument shows that
$d_{m,n}=\sum_{x+y=m}\big{(}\sum_{z+w=n}c_{x,z}^{1}c_{y,w}^{2}\big{)}$. This
is precisely what arises when the polynomials are multiplied.

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