A Formula for the HOMFLY Polynomial of rational links

Rational (or 2-bridge) knots and links constitute an important class of links for which many problems of knot theory can be completely solved and provide examples often leading to general theorems about arbitrary knots and links. For the basics on rational (2-bridge) knots and links we refer the reader to [Lickorish1997] and [Murasugi1996]. As regards the definition, we follow [Lickorish1997], while the majority of properties that we need, are to be found in a more detailed exposition of [Murasugi1996]. In particular, by equivalence of (oriented) links $L=K_{1}\cup K_{2}$ and $L^{\prime}=K_{1}^{\prime}\cup K_{2}^{\prime}$ we understand a smooth isotopy of $\R^{3}$ which takes the union $K_{1}\cup K_{2}$ into the union $K_{1}\cup K_{2}$, possibly interchanging the components of the link.

Let $p$ and $q$ be mutually prime integers, $q>0$, $\left|{p\over q}\right|\leq 1$, and we have a continued fraction

where $b_{i}$ are nonzero integers (positive or negative). Below, we will use shorthand notation $[b_{1},b_{2},\ldots,b_{n}]$ for the continued fraction with denominators $b_{1},b_{2},\ldots,b_{n}$. A theorem of Schubert (see, for instance, [Lickorish1997]; [Murasugi1996]) says that the (isotopy type of the) resulting unoriented link does not depend on the choice of the continued fraction for the given number $p/q$.

The case $p=q=1$ is exceptional: it corresponds to the trivial knot which is the only rational, but not 2-bridge knot. On some occasions, it will be helpful to allow the numbers $b_{i}$ also take values 0 and $\infty$ subject to the rules $1/0=\infty$, $1/\infty=0$, $\infty+x=\infty$.

Consider a braid on four strands corresponding to the word $A^{b_{1}}B^{b_{2}}A^{b_{3}}\ldots$, where $A$ and $B$ are fragments depicted in Fig. 1 and concatenated from left to right.

Then take the closure of this braid depending on the parity of $n$ (see Fig. 2).

We will call (non-oriented) diagrams obtained in this way natural diagrams of rational links and denote them by $D[b_{1},b_{2},\ldots,b_{n}]$. We shall denote the link represented by this diagram as $L\left({p\over q}\right)$. For odd denominators $L\left({p\over q}\right)$ turns out to be a knot, while for even denominators it is a two-component link. Such knots and links are called 2-bridge or rational.

Note that, if a natural diagram represents a two-component link, then the two vertical leftmost fragments belong to different components. If they are oriented in the same direction, as shown in Fig. 4, then we call the diagram positive and denote it by $D^{+}[b_{1},b_{2},\ldots,b_{n}]$.

If the orientation of one of the components is reversed, then we call it negative and denote by $D^{-}[b_{1},b_{2},\ldots,b_{n}]$. It does not matter which component of the link is reversed, because the change of orientation of both components yields the same link, see [Murasugi1996]. As we will see later (Lemmas 3, 4), the choice between the corresponding links does not depend on a particular continued fraction expansion of the number $p/q$. This makes the notations $L^{+}(p/q)$ and $L^{-}(p/q)$ well-defined.

Let $p^{\prime}=p-q$, if $p>0$, and $p^{\prime}=p+q$, if $p<0.$ According to [Murasugi1996], we have: $L^{-}\left({p\over q}\right)=L^{+}({p^{\prime}\over q})$, therefore, in principle, it is sufficient to study only the totality of all positive rational links. In the case of knots (when $q$ is odd), the two oppositely oriented knots are isotopic, and we have $L\left({p\over q}\right)=L\left({p^{\prime}\over q}\right)$ (again, see [Murasugi1996]). Therefore, it is sufficient to study only the knots with an even numerator (cf. Lemma 2 below).

Another important operation on links is the reflection in space; it corresponds to the change of sign of the corresponding rational number: $p/q\mapsto-p/q$, see [Murasugi1996].

The two symmetry operations on rational links generate a group $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$; they are transparently exemplified by the examples $p/q=1/4,-1/4,3/4,-3/4$, which correspond to the four versions of the so called Solomon knot (although it is actually a two-component link):

By dragging the lower strand of the diagram for $L(3/4)$ upwards we get the diagram for $L(-1/4)$ with the opposite orientation of the upper strand. The same is true for the pair $L(-3/4)$ and $L(1/4)$.

In 2004–2005 Japanese mathematicians [Fukuhara2005] and [Mizuma2004] found independently different explicit formulae for the simplest invariant polynomial of 2-bridge links: the Conway (Alexander) polynomial. The aim of the present paper is to establish a formula for a more general HOMFLY polynomial $P$ in terms of the number $p/q$ that defines the rational link.

The HOMFLY polynomial ([Lickorish1997]; [Prasolov and Sossinsky1997]; [Chmutov et al.2012]) is a Laurent polynomial in two variables $a$ and $z$ uniquely defined by the following relations (we use the normalization of [Knot Atlas2015]; [Chmutov et al.2012]; other authors may write the same polynomial in different pairs of variables, for example, [Lickorish1997] uses $l=\sqrt{-1}a$ and $m=-\sqrt{-1}z$):

where $L_{+}$, $L_{-}$ and $L_{0}$ are links that differ inside a certain ball as shown in Fig. 5.

As we mentioned in the previous section, in the case of rational knots, the change of orientation gives the same (isotopic) knot, while for links it is important to distinguish between the two essentially different orientations (this number is two, not four, because the change of orientation on both components gives the same rational link).

There is a simple formula relating the HOMFLY polynomials of a knot (link) with that of its mirror reflection ($a\mapsto-a^{-1}$, $z\mapsto z$), so in principle it is enough to study only the knots (links) described by positive fractions.

HOMFLY polynomials of some links are given below in Fig. 7 and Table 1.

Consider a family of links $L_{n}$ for $n$ even, which coincide everywhere but in a certain ball, where they look as shown in Fig. 6a–c. Moreover, we define the link $L_{\infty}$ by Fig. 6d. That is, we consider a family of links with a distinguished block where the strands are counter-directed. A formula similar to what we are going to prove, can also be established for co-directed strands, but for our purposes the following Proposition is sufficient. It expresses the value $P(L_{n})$ through $P(L_{0})$ and $P(L_{\infty})$.

Particular cases of this Corollary for $n=0,\ \pm 2,\ \pm 4$ give the well-known values of the HOMFLY polynomial for the two unlinked circles, the Hopf link and the two (out of the total four) versions of the oriented “Solomon knot”, see Fig. 7.

The continued fraction expansion with even denominators and the corresponding natural diagram will be referred to as the canonical expansion of a rational number and the canonical diagram of a rational link (defined up to a rotation, see Lemma 4).

Now we are in a position to define a canonical oriented rational link.

Let

where $q$ and all $b_{i}$ are even. The diagram $D[b_{1},\ldots,b_{n}]$ taken with the positive orientation, denoted by $D^{+}[b_{1},\ldots,b_{n}]$, will be referred to as the canonical diagram of the oriented link $L^{+}(p/q)$.

We will use the canonical diagrams for the proof of the main theorem. However, for this theorem to make sense, we must check that the oriented link $L^{+}(p/q)$ does not depend on a particular choice of the continued fraction for the rational number $p/q$ and, especially, that it does not change when $p/q$ is changed by $\bar{p}/q$ where $p\bar{p}\equiv 1\mod 2q$. We will prove these facts immediately.

The previous lemma justifies the notation $L^{+}(p/q)$.

In a canonical diagram of a rational link, due to the fact that all blocks are of even length, the strands are everywhere counter-directed. Therefore, Lemma 1 can be applied recursively:

because the following two pairs of diagrams are equivalent as links: $D^{+}[b_{1},\ldots,{}b_{n-1},0]=D^{+}[b_{1},\ldots,b_{n-2}]$ and $D^{+}[b_{1},\ldots,b_{n-1},\infty]=D^{+}[b_{1},\ldots,b_{n-1}]$. The sign $\varepsilon_{n}=(-1)^{n-1}$ comes from our convention of counting the number of twists in the first and the second layers of a natural diagram (see Fig. 1—the powers of $A$ and $B$ correspond to odd and even values of $n$, respectively).

For a given sequence $[b_{1},\ldots,b_{n}]$ denote $x_{n}=P(D^{+}[b_{1},\ldots,b_{n}])$. Then Eq. (5) can be rewritten as

which makes sense when $n>2$. Drawing the diagrams and applying skein relation (2) for the cases $n=2$ and $n=1$, we can see that Eq. (4) still holds for these values, if we set $x_{0}=1$ and $x_{-1}=z^{-1}(a-a^{-1})$.

Our aim is to find a closed form formula for $x_{n}$ in terms of $a$ and $z$. To do this, it is convenient to first consider a more general situation.

To obtain the formula for the HOMFLY polynomial of an arbitrary rational link $L^{\pm}(p/q)$, we combine Lemma 5 with formula (4). For the sake of unification, we first make some preparations:

• •

  If $q$ is odd (that is, we deal with a knot) and $p$ is odd, too, then we change $p$ to $p^{\prime}=p-q$, if $p>0$, or to $p^{\prime}=p+q$, if $p<0$. Then $L(p/q)=L(p^{\prime}/q)$, the numerator of the fraction becomes even, hence Lemma 2 applies and formula (4) is valid.

• •

  If $q$ is even and the link is negative, then we use the property $L^{-}(p/q)=L^{+}(p^{\prime}/q)$, where $p^{\prime}$ is computed by the same rule as above. Below, we will simply write $L(p/q)$ instead of $L^{+}(p/q)$.

Now the main result reads:

Then formula (5) gives:

In formula (5) one can, in principle, collect the terms with equal powers of $z$. The formulation of this result is rather involved, and we need first to introduce necessary notations.

Let $\alpha=p/q$ be a nonzero irreducible rational number between $-1$ and 1. We denote by $n=\nu(\alpha)$ the length of the canonical continuous fraction for $\alpha$, and by $\alpha^{\prime}$, the number $\alpha+1$, if $\alpha<0$, and $\alpha-1$, if $\alpha>0$. Now, let

where $\overline{1,n}=\{1,2,\ldots,n\}$ and for $S\subset{\overline{1,n}}$ the set $S-1$ is understood as the set of all numbers $m-1$, where $m\in S$.

Then we have:

The theorem can be proved by first collecting the terms with equal powers of $z$ in the statement of Lemma 5 and then using induction on $\nu(\alpha)$; we do not give the details here. Although Theorem 2 is in a sense more explicit than Theorem 1, it is less practical; in particular, the formula of Theorem 1 is better suited for programming purposes.

The formula for $P(L(p/q))$ can be easily programmed. The source code of the program, written by the second author and tested by the first one, as well as the resulting table of HOMFLY polynomials for rational links with denominators not exceeding 1000, are presented online at [Computer generated table of HOMFLY polynomials for rational links2010]. Below, we give a short excerpt of that big table which is enough to know the polynomials of all rational knots and links with denominators no ${<}9$, if one uses the following rules (see [Murasugi1996]):

1. (1)

   $P(L^{+}(-p/q))$ is obtained from $P(L^{+}(p/q))$ by the substitution $a\mapsto-a^{-1}$.

2. (2)

   The knots $L(p_{1}/q)$ and $L(p_{2}/q)$ are equivalent, if $p_{1}p_{2}\equiv 1\mod q$.

3. (3)

   The links $L^{+}(p_{1}/q)$ and $L^{+}(p_{2}/q)$ are oriented equivalent, if $p_{1}p_{2}\equiv 1\mod 2q$.

In Table 1, the first column (R) gives the notation of the rational link (knot) as $L(p/q)$ (in the case of links, this means $L^{+}(p/q)$), the second column (T) contains the standard notation of that link (knot) from Thistlethwaite (Rolfsen) tables (see [Knot Atlas2015]; the bar over a symbol means mirror reflection, the star is for the change of orientation of one component), and the third column (H) is for the values of the HOMFLY polynomial. Note that we list HOMFLY polynomials for both orientations of each rational link, while the famous [Knot Atlas2015] shows them for only one orientation of two-component links.

1. 1.

   As the Conway polynomial is a reduction of the HOMFLY polynomial, Theorem 1 gives a formula for the Conway polynomial of rational links by the substitutions $a=1$, $z=t$ [the fraction $(1-a^{n})/(a-a^{-1})$ is first transformed to a Laurent polynomial and becomes equal to $-n/2$].

2. 2.

   Since the Jones polynomial is a reduction of the HOMFLY polynomial, Theorem 1 leads to a formula for the Jones polynomial of rational links by the substitutions $a=t^{-1}$, $z=t^{1/2}-t^{-1/2}$.

3. 3.

   The famous open problem whether a knot must be trivial if its Jones polynomial is 1, has a simple positive solution for rational knots. Indeed, the value $|J(-1)|$ is equal to the determinant of the knot, and the determinant of a rational knot is its denominator (see [Murasugi1996]).

4. 4.

   Open problem Can one generalize formula (5) to all links? The results of [Traldi1989] show that it can be generalized to at least some non-rational links, although his formula is less explicit than ours.