数理解析研究所講究録
第 1911 巻 2014 年 101-112

101

Recent advances on 1-cocycles in the space of knots
Arnaud Mortier
OCAMI, Osaka City University
Abstract

This is a survey of two recent papers [8, 13] in which were introduced new methods for constructing
1-cocycles in the space of knots. The construction from [13] is a natural adaptation of Polyak-Viro’s
formulas for finite-type knot invariants; it is conjectured to give the first combinatorial formulas forvalued Vassiliev 1-cocycles. In [8], the cocycles take values in skein modules associated with quantum
knot invariants; conjecturally, the examples produced detect information regarding the geometry ofknots.
For a broader panorama on this topic, see also [6, 7, 15, 16, 19], Sections 1.6-1.8 of [5], and [9]which is a sequel to [8],$\mathbb{Z}-$1

Finite-type 1-cocycles of knots given by Polyak-Viro formulas[13]

Vassiliev’s cohomology classes were introduced in 1990, at a time when the cohomology ofthe space of knots had barely been studied. In the years that followed, two kinds of explicit
formulas were proved to describe all of Vassiliev’s -cocycles, best known as finite-typeinvariants: an integral formula, due to Kontsevich [12], and purely combinatorial formulas
due to Polyak-Viro [14] (see also [10]).
In the meantime, in higher degree, only one example was proved to exist, by Teiblum
and Turchin (at the time a student of Vassiliev), and this 1-cocycle
waited unti12001
before Vassiliev [19] found how to actually evaluate it, over
, with a combinatorial
formula involving diﬀerential geometry. Ten years later, Sakai [15] described a realizationof over by means of an integral formula.
The purpose of [13] is to fill the gap of a missing combinatorial formula forover
to remove geometric conditions from the computation process, and to find more examplesof 1-cocycles. Just like the original works of Vassiliev [17, 18], this article considers thespace of smooth long knots–i.e., embeddingsthat are standard outside of , 1$0$$v_{3}^{1}$$\mathbb{Z}_{2}$$v_{3}^{1}$$\mathbb{R}$$v_{3}^{1}$$\mathbb{R}\mapsto \mathbb{R}^{3}$

1.1$\mathbb{Z},$

$[0$

$].$

Preliminary: Gauss diagram formulas (after [14])The spirit of Polyak-Viro’s Gauss diagram formulas is to count the subdiagrams of a knot,
with weights. Here, knot diagrams are represented combinatorially using Gauss diagrams-see Fig.1, and a subdiagram is the result of removing (possibly empty) set of arrows.
One of the origins of that idea lies in the well-known formula for computing the linking
number of a 2-component oriented link: given a link diagram, $lk(L_{1}, L_{2})$ is the sum of
writhes of all crossings wheregoes over. In Polyak-Viro’s language, each such$a$$L_{1}$$L_{2}$

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crossing is a subdiagram with only one arrow remaining, oriented fromto , and theweight given to each subdigram is the writhe of the crossing.
In [14], Polyak and Viro give such formulas for computing (among others) the first two
Vassiliev invariants (Theorems 1 and 2). In both cases (as well as in the linking number
formula), the weight given to a subdiagram is equal to the product of its writhe numbers,
times a constant which depends only on the underlying unsigned diagram. Such particular
choices of weights proved to be extremely common in further literature (see for instance[3, 2, 4 and yield what is often called arrow diagram formulas (an arrow diagram is a
Gauss diagram deprived from its decorating signs).
$L_{1}$$\nearrow^{\cross+\backslash }$$L_{2}$$/_{-}\backslash ^{\nearrow}$$\sim$$\cross^{\backslash }$Figure 1: A long figure eight knot diagram and its Gauss diagram

Figure 2: Example of computation of the Casson invariant on a random Gauss diagram

The main interest of these formulas lies in the following result.

Theorem 1 (Polyak-Viro [14], Goussarov [10]). A knot invariant admits a Gauss diagram

formula if and only if it is a Vassiliev invariant.
Although the proof given by the authors of this theorem does not mention the original
definition of Vassiliev invariants, it is possible to obtain such formulas by computing
weight systems (see [1]) and integrating them via the homological calculus presented in[19]. This process highlights the deep origin of the writhe numbers in these formulas, as
co-orientations of singular strata in the space of all (including singular) knots. It is also
one reason to believe that computing products of writhes at a higher level could yieldformulas for Vassiliev 1-cocycles.
1.2

Arrow germ formulas

The idea in [13] is to copy Polyak-Viro’s construction using as raw material not a Gauss
diagram, but a Reidemeister move, which is here regarded as an elementary path in thespace of knots.

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Definition 1. An -germ $(i=1,2,3)$ is a couple of Gauss diagrams that diﬀer only by a
Reidemeister move.
partia13-germ is a 3-ge7m with one arrow removed (in both diagrams) from theReidemeister triple.
subgerm is the result of removing a set of arrows from (possibly partial) germ, consistently in both diagrams. Arrows involved in the Reidemeister move cannot be removed,
except in 3-germs where at most one arrow from the triple can be forgotten.
As before, these notions have a counterpart with no sign decorations, called (partial)
arrow $germs-and$ as before, the latter are meant to count subgerms, weighted with the
product of writhes of the arrows involved.
The main result of [13] is the following.
Theorem 2. The formal sumof (partial) arrow germs on Fig.3 defines 1-cocyclein the space of long knots, over . Its reduction mod 2 coincides with Teiblum-Turchin’s
$i$

$i$

$A$

$A$

$a$

$\alpha_{3}^{1}$$a$$\mathbb{Z}$cocycle$v_{3}^{1}.$The second statement is proved by showing directly thatmod2 is “of finite type”,
using Vassiliev’s homological calculus presented in [19]. The proof of the first statement relies on a higher order Reidemeister theorem, that is, an exhaustive list of all
2-codimensional strata corresponding to the most generic degeneracies of Reidemeistermoves (by definition, a 1-cocycle should vanish on the meridians of such strata). A complete description of those strata and their meridians is given in [13]; see also [7].
$\alpha_{3}^{1}$

Notation:

Figure 3: The first non-trivial arrow germ formula$\alpha_{3}^{1}$: three partial arrow 3-germs and one arrow 3-germ

As one can see, no 1- or 2-germs are involved in the formula of Theorem 2, and thisis actually a general fact: in any cohomology class represented by germ formulas, there
is a formula which contains only (possibly partial) 3-germs ([13], Proposition 2.8). As aresult, the linear system with germs as variables and 2-meridians as equations is reduced
to a reasonable size ([13], Theorem 2.11).
Conjecture 1. Every 1-cohomology class with an arrow germ presentation is of finite
type in the sense of Vassiliev.
So far, only one way is known for proving that a 1-cocycle is of finite-type over ;it consists in defining orientations on the varieties involved in Vassiliev’s homologicalcalculus [19]. It has not been done yet, to the best of our knowledge.
$\mathbb{Z}$

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1.3

Evaluation of$\alpha_{3}^{1}$

on canonical cycles

One interest of 1-cocycles is that evaluating them on loops that are defined canonicallyfor all knots produces knot invariants. The easiest example of such a loop, for long knots,
is the loop rot (K) which consists of a full positive rotation of $K$ around its axis–see
Figs.4 and 5. It is proved in [13] thatis equal to minus the Casson invariant$K$of :$\alpha_{3}^{1}(rot(K))=-v_{2}(K)$ .
$\alpha_{3}^{1}(rot(K))$

This equality was conjectured to hold for the Teiblum-Turchin cocycle in [16]. Turchin’s
conjecture would follow from the above result and Conjecture 1.

Figure 4: One realization of the loop rot (K) as a sequence of Reidemeister moves

Figure 5: The railway followed by $K$ on Fig.4 is isotopic to a full rotation around the axis

A result of Hatcher [11] states that besides rot (K) , there is essentially only one otherinteresting loop in the moduli space of all long knots equivalent to a given prime knot.
This second loop, often called the Hatcher loop Hat (K) , consists of sliding alittle “ball at
infinity”’ along a fixed parametrization of the knot in(a framing convention should be
made, because every time the ball makes a full spin around itself, it adds $\pm rot(K)$ to theloop). Although it does not seem easy to evaluate
on Hat (K) , further investigationsoshowed that there is another arrow germ formula , with the same properties asfar, and such that if $K$ has framing , then$\mathbb{S}^{3}$$\alpha_{3}^{1}$$\tilde{\alpha}_{3}^{1}$$\alpha_{3}^{1}$$0$$\tilde{\alpha}_{3}^{1}$(Hat (K) ) $=-6v_{3}(K)$ .

Here denotes the only Vassiliev invariant of order 3 with value 1 on the positive trefoil,$-1$ on the negative trefoil andon the unknot. This result is to appear in a subsequent
article.
$v_{3}$$0$105
Quantum one-cocycles for knots [8]2

In this article, Fiedler defines a new family of 1-cocycles in the topological moduli spaceof long knots, which are then extended to cocycles in the space of string links. As in theprevious article (Section 1), the formulas are constructed as algebraic intersection forms
with the variety of Reidemeister moves, purely combinatorially: follow a loop in the spaceof knot diagrams, and every time you meet a Reidemeister move, count something. Only
this time what you count does not live in a “small” group like , but in a more complicatedstructure that stays closer to topology and keeps more information.
One of the motivations here is to detect geometry-related information, with the following result in mind (see the definitions of rot and Hat in the previous section).
$\mathbb{Z}$Theorem 3 (Hatcher [11]). Let $K$ be a long knot which is not a satellite. Then the loops
rot (K) and Hat (K) are linearly independent over if and only if $K$ is hyperbolic.
$\mathbb{Q}$It means that if one can create a 1-cocycle powerful enough to detect exactly whenrot (K) and Hat (K) are linearly dependent, then it gives a simple criterion to evaluate thegeometry of prime knots. The ultimate goal in that direction is to find a cocycle suchthat the quotient v(rot)/v(Hat), when not a rational number, is related to the hyperbolic
volume.
$v$

The space of cochains

2.1

There are a lot of combinatorial tools required to define properly the objects in Fiedler’s
theory; we begin with a simplified version, which will be refined gradually in the next
subsections. In particular, we first set aside orientation and sign issues.
Let us describe the elementary cochains from which one tries to obtain 1-cocycles.
They are based on a simple surgery operation that generalizes knot mutations. Indeed,
let be a tangle diagram in a 2-disc, with 6 boundary points. One defines an elementarycochainas follows. Pick a generic path in the space of long knot diagrams. For
each Reidemeister III move involved in , notice that the local Reidemeister picture is atangle diagram in a disc, with 6 boundary points; remove that disc from your diagram,and replace it with , making sure that the boundary points match; there are six ways
to do that, and we will explain later how to choose one canonically. The evaluation
is by definition the formal sum of all tangle diagrams so obtained.
By considering tangles with 4 (respectively, 2) boundary points, one can similarly
construct cochains that detect Reidemeister II (respectively, I) moves.
Now if one stops here, the space of cochains is indexed by tangles in a disc, so that it isinfinite-dimensional. Moreover, a brief examination reveals that there are very few chances
to ever find a cocycle in that space: indeed, a 1-cocycle should vanish on 2-meridians,
and it is not easy for a formal sum of tangle diagrams to vanish (even with proper signs
defined). The natural answer to both these issues is to push the values of the cochainsin a skein module. In [8], two modules are separately considered: the HOMFLYPT and
Kauﬀman polynomials, which both yield finitely generated cochain spaces. Note that in
the former case, all graft tangles should be oriented and surgeries performed consistently.
$T$$c_{T}$$\gamma$$\gamma$$T$$c_{T}(\gamma)$

$T$

106

へ$\sim$$c_{T}(\gamma)=$Figure 6: Elementary cochain associated with an oriented tangle
in Subsection 2.1.3.

$T$. The middle strand issue is explained

Local and global types

2.1.1

An empirical fact in this theory is that 1-cocycle formulas tend to contain a lot of symmetry, and the more symmetry there is in a formula, the more likely it is to be cohomologicallytrivial. There are mainly two ideas in [8] to get round this diﬃculty: first, use parametersand decorations so as to break the symmetry as much as possible; second, find a way to
extract interesting information even from a cocycle that is cohomologically trivial (seeSection 2.2.1).
Let us assume for now that we work with long knot diagrams (see example on Fig.1).
There are a number of data that one can read given a Reidemeister move in such adiagram, which together define the local and global types of the move. Then, if denotes
one particular local type, and a global type, one can enrich the previous settings by
but only if: it makes the same computation as
defining an elementary cochain$(l,g)$
over all types
is the sum of(so that
the Reidemeister move has type
and .
Local types are more conveniently read on the local knot diagram picture, while global
types require to look at how the involved crossings are arranged on the Gauss diagram,with respect to each other and to the point at infinity; the point at infinity appears to be
the key to break the symmetry and allow non-trivial solutions to exist.
Figure 7 sums up the definitions of all types, and where to find them in [8].
$l$

$g$

$c_{T}$$c_{T,l,g}$$l$$c_{T}$$c_{T,l,g}$$g)$, although convenient for an overview of the article,
Terminology. The notation
is not the one used in [8]. There, cochains are systematically defined by giving, for eachcouple of types $(l, g)$ , which tangles (called partial smoothings) contribute. For instance,means in the current
for type 3 is“for the cochain , the partial smoothingterminology that can be written”$c_{T,l,g}$

$T$

$R$

$T_{r_{c}}$

$R$

$R=c$

3, $r_{C}+$other terms.

107

Figure 7: All local and global types with their reference pages in [8] (version 2)

2.1.2

Signs

One obvious necessary condition for a cochain to define a 1-cocycle is that should
vanish on any little loop in which one Reidemeister move is performed and then undone.
As a result, one must either work over
, or define “co-orientations” of Reidemeistermoves. The latter choice is made in [8], as follows.
In the case of Reidemeister I and II moves, the evaluation of an elementary cochain
comes with an additional minus sign if and only if the move destroys crossings.
In the case of Reidemeister III, the co-orientation is defined separately for each local
type. That is the meaning of $the+and-$ signs in [8] Fig.16 p.31, they indicate on whichside of the move each picture is. When evaluating a cochain, count an additional minus
sign if and only if the move goes from the positive to the negative side. Note that thesigns here were not chosen arbitrarily. The easiest way to show it might be to point out
that in Fig.8, where every local type appears exactly once, all eight triangles are on theirrespective positive side.
$c$$\mathbb{Z}/2$$c_{T,l,g}$

$c$

108
1

Figure 8: All eight local types on their positive side, in one picture
2.1.3

Gluing conventions

We now explain how Fiedler chooses one out of six ways of gluing a graft tangle after
removing a Reidemeister III tangle (for I there is no choice since the skein modules are
1-dimensional, and for R II it is directly made clear on each picture, like [8] Fig.41 p.59,
which in our current notations would read $-(v-v^{-1})c$ ) $(,+,0)$ .
First, notice that for local types 2 and 6 (called star-like), sources and sinks alternate
along the boundary of the Reidemeister tangle; for the remaining (braid-like) types 1,
3, 4, 5, 7 and 8, the sources can be grouped together on one side of the disk. In thelatter case, the convention is that the three boundary points at the left of any picture ofa graft tangle should replace the three sources of the removed tangle. This convention
is especially useful when working over an unoriented skein module; otherwise, the simple
fact that orientations should match leaves no choice.
For local types 2 and 6, one defines the mid point in the Reidemeister tangle as
the source of the “middle strand, which goes neither over the two other strands, norunder them. The convention is then to decorate one of the boundary points in the graft
tangles with a letter $m$ ” , indicating that this point always replaces the mid point when
performing a surgery–see Fig.6.
When working over an oriented skein module, only graft tangles with consistent orientations are allowed; note that this consistency condition depends on the local type.
$R$

$T$

2.1.4

Weights

There are two major ingredients in Fiedler’s formulas. The first is the idea of performing
surgeries that formally depend on some local and global parameters, i.e., the cochains
defined so far. The second is to weight these cochains with Gauss diagram
formulas. Very roughly, it amounts to considering the module freely generated by the, not over , but over a ring of arrow diagrams (as defined inelementary cochains
Section 1).
As in Section 1, those arrow diagrams compute sums of products of writhes in the Gauss
diagram that represents the long knot at the moment where each Reidemeister move isperformed. The number of writhes involved in each product determines the degree of theweight. A weight of degree 1 (resp. 2) is called linear (resp. quadratic).
$c_{T,l,g}$$\mathbb{Z}$$c_{T,l,g}$

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2.2

Results

Recall that when a 1-cochain in the space of knots is defined as an intersection form with
Reidemeister moves, the condition for being a cocycle is to vanish on the meridians of the2-codimensional strata defined by the higher order Reidemeister theorem (just like knot
invariants, i.e., -cocycles, should vanish on the meridians of the 1-codimensional strata
defined by the usual Reidemeister theorem). Two of these strata are by far the mostcomplicated, and occupy a central place in [8]:$0$$\bullet$$\bullet$The set of knots whose projection to the plane contains a quadruple point (which
can be thought of as an arc sliding over a Reidemeister III move), denoted $by*.$The equations associated with their meridians are called the tetrahedron equations.

The set of knots whose projection contains a triple point with two tangent branches(an arc sliding over/under/through a Reidemeister II move), denoted by . The
associated equations are called the cube equations.
$\succ|\prec$

The results are organized as follows: for each skein module (HOMFLYPT, pp.34-149
and Kauﬀman, pp.149-168), two 1-cocycle formulas are constructed:
andin
the HOMFLYPT case,andin the Kauﬀman case. Each of these are built
step by step, by first solving the tetrahedron equation, then adjusting the solution sothat it satisfies also the cube equations, then adjusting again so as to get all (whenpossible) remaining equations satisfied. Formulas whose name contains a subscript “reg”
do not vanish on meridians involving Reidemeister I moves; however they vanish on allother meridians, which makes them regular cocycles, i,e., invariants of regular loops up
to regular isotopy ([8] Theorem 3 p.94); they are both made of quantum cochains withlinear weights. On the other hand,satisfy all equations and define non-trivial
andcocycles in the space of long knots ([8] Theorem 4 p.127 and Theorem 5 p.165). They are
made of a main part which is a quantum cochain with (at most) quadratic weights, anda corrective term which purely consists of a weight of degree 3 (and no surgery part).
$R_{\eta eg}^{(1)}$$R_{F,reg}^{(1)}$$R_{F}^{-(1)}$$R^{(1)}-$2.2.1$\overline{R}^{(1)}$$R_{F}^{-(1)}$From long knots to string links

There are many reasons for which one would like to generalize a construction related tolong knots to string links in a 3-ball. For instance, producing families of invariants via
knot cabling, or generalizing a formula computing the trivial knot invariant, which may
prove to be non-trivial when applied to string links. When it comes to 1-cocycles, tangles
in general are a little bit unfriendly: the moduli space of all tangles isotopic to a given one
has very often a trivial first homology group. However, even a 1-cocycle formula that isnull-cohomologous can lead to non-trivial invariants of tangles if evaluated on a canonicalarc which is not a loop.
The way Fiedler extends his long knot cocycle formulas to arbitrary string links isextremely simple: just like knots, string links can be represented with Gauss diagrams,
the only diﬀerence is that the base manifold is not a circle, but a collection of numberedarcs. Fix arbitrarily:

1. a way of gluing these arcs together so that they form a circle;

110
2. a point at infinity among the gluing points.
That is all. Now any sequence of Reidemeister moves applied to any string link diagram
looks like a sequence applied to a long knot, and it makes sense to evaluate Fiedler’s
formulas there. Any quantum cocycle for long knots defines a family of cocycles in themoduli space of string links, indexed by the above choices of a circular permutation anda point at infinity.
2.2.2

The scan-arc and the scan-property

As mentioned earlier, there are no non-trivial loops in the moduli space of string links ingeneral, except for particular cases such as long knots and their cables (for which there isa version of the Hatcher $1oop-[8]$ , p.8).
Fiedler introduces a scan-arc, defined canonically for all string links, and denoted byscan (L) . We represent it on Fig.9. It can be thought of as a generalization of the looprot (K) defined for long knots: indeed, rot (K) consists basically of two consecutive scans”(Fig.4). Notice that the little loop on the first frame is not a part of the string link, andits “creation” by Reidemeister I is not a part of the scan-arc either (although the sequenceof Reidemeister II moves that lead to the second frame is included in the scan-arc): it
follows that the scan-arc is regular (it does not involve any Reidemeister I move), hence
it makes sense to evaluate
andthere.
$\langle$$R_{veg}^{(1)}$$R_{F,reg}^{(1)}$$arrow$$arrow$

Figure 9: The scan-arc of a string link

Now one would like a result of the type “Let$L$

be a quantum cocycle formula. Then
It is not diﬃcult to see that thisholds true if is a quantum formula with constant weights. However it fails with arbitrary$c$$c(scan(L))$ is an isotopy invariant of the string link L.”$c$weights.
By definition, a quantum cocycle formula is said to have the scan-property if $c(scan(L))$
defines an isotopy invariant of the string link . All four formulas constructed in [8] havethat property ([8] Theorems 3, 4, 5).
Among many examples, let us mention that if is chosen to be the positive generator of
the 2-strand braid group, and if the point at infinity is chosen correctly, then $R_{7eg}^{(1)}(scan(L))$
is equal to the HOMFLYPT polynomial of the right trefoil (times the class of the trivial
braid in the HOMFLYPT skein module).
, the computations made lead to the following conjectures ([8], Conjectures
As for1 and 2).
$c$

$L$

$L$

$\overline{R}^{(1)}$Conjecture 2. Let $K$ be a long knot, let $v_{2}(K)$ be its Casson invariant and let$P_{K}$be111
its HOMFLYPT polynomial. Let
component link. Then$\delta$$\overline{R}^{(1)}$denote the HOMFLYPT polynomial of the trivial 2-(rot (K) ) $=\delta v_{2}(K)P_{K}.$Conjecture 3. Let $K$ be a long knot with trivial framing and with non trivial Casson
invariant. Then $R^{(1)}(Hat(K))$ is a non zero integer multiple of $R^{(1)}(rot(K))-$ if and only
if $K$ is a torus knot.
So far those two conjectures have been checked for knots up to 4 crossings.
2.2.3

Gradings

One specificity of the formulas
andextended to string links is that they can)be refined into a collection of cocycle formulas
and $Rreg$(A) indexed by a set of
gradings, so that the original formulas are the sum of the refined formulas over all possible$R_{eg}^{(1)}$$R_{F,reg}^{(1)}$$R_{\tau eg}^{(1)}(A)$gradings.
A grading is defined for any Reidemeister II or III move. Just like
is oblivious ofReidemeister moves that are not of type $(l, g)$ ,anddisregard the moves
that do not have grading . The full definition, however, is not very enlightening, so we
do not mention it here; see [8], Definition 9 p.48.
$c_{T,l,g}$$R_{\gamma eg}^{(1)}(A)$$R_{Fre}^{(1)}(A)$$A$

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OCAMI, Osaka City University
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka-shi
Osaka 558-8585, JAPAN-mail address: arno.mortier@laposte.net$E$