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Two graph homologies and the space of long embeddings

Leo Yoshioka Graduate School of Mathematical Sciences, The University of Tokyo
  e-mail:[email protected]
(October 2023)
Abstract

Graph homologies are powerful tools to compute the rational homotopy group of the space of long embeddings π∗​Emb​(ℝj,ℝn)⊗ℚ\pi_{\ast}\text{Emb}(\mathbb{R}^{j},\mathbb{R}^{n})\otimes\mathbb{Q}. Two graph homologies have been invented from two approaches to study the space of long embeddings: the hairy graph homology from embedding calculus, and BCR graph homology from configuration space integral. In this paper, we construct a monomorphism from the top hairy graph homology to the top BCR graph homology, though the latter graph homology is quite modified. This map and its left inverse are analogs of PBW isomorphism between ℬ\mathcal{B} and 𝒜​(S1)\mathcal{A}(S^{1}), the space of open Jacobi diagrams and the space of Jacobi diagrams on the unit circle, in the theory of Vassiliev knot invariants.


Introduction

A long embedding is an embedding of ℝj\mathbb{R}^{j} into ℝn\mathbb{R}^{n}, which is standard outside a disk in ℝj\mathbb{R}^{j}. We write 𝒦n,j\mathcal{K}_{n,j} for the space of long embeddings Emb​(ℝj,ℝn)\text{Emb}(\mathbb{R}^{j},\mathbb{R}^{n}), and write 𝒦¯n,j\overline{\mathcal{K}}_{n,j} for the homotopy fiber of the forgetful map 𝒦n,j→Imm​(ℝj,ℝn)\mathcal{K}_{n,j}\rightarrow\text{Imm}(\mathbb{R}^{j},\mathbb{R}^{n}) to the space of long immersions. The aim of this paper is to compare two combinatorial objects that are introduced to study 𝒦¯n,j\overline{\mathcal{K}}_{n,j}.

In 2017, Fresse, Turchin and Willwacher [FTW], following Arone and Turchin [AT1][AT2], established a beautiful framework to compute the rational homotopy group π∗​𝒦¯n,j⊗ℚ\pi_{\ast}\overline{\mathcal{K}}_{n,j}\otimes\mathbb{Q} in terms of combinatorial object called graph homology. More precisely, they showed the isomorphism between π∗​(𝒦¯n,j)⊗ℚ\pi_{\ast}(\overline{\mathcal{K}}_{n,j})\otimes\mathbb{Q} and the graph homology H∗​(H​G​Cn,j)H_{\ast}(HGC^{n,j}), called hairy graph homology. This significant result is based on the theory of embedding calculus established by Goodwillie, Klein and Weiss [GW][GKW][Wei] and also the theory of little disk operad.

However, this deep framework is fully successful, so far, only for the range n−j≥3n-j\geq 3. Moreover, it seems difficult to know geometric generators of π∗​(𝒦¯n,j,ℚ)\pi_{\ast}(\overline{\mathcal{K}}_{n,j},\mathbb{Q}) from generators of H∗​(H​G​C)H_{\ast}(HGC).

On the other hand, from 90’s to 10’s, Bott[Bot], Cattaneo, Rossi[CR], Sakai[Sak] and Watanabe[Wat] developed a framework to construct geometric (co)cycles of 𝒦¯n,j\overline{\mathcal{K}}_{n,j} from another graph (co)homology. We call their graph homology BCR graph homology and write it as H∗​(G​Cn,j)H_{\ast}(GC^{n,j}). To get the cocycles of 𝒦¯n,j\overline{\mathcal{K}}_{n,j}, they used the operation called configuration space integral. Fortunately, this framework is applicable to n−j=2n-j=2.

Surprisingly, BCR graph homology H∗​(G​C)H_{\ast}(GC) has very similar graphs to those of H∗​(H​G​C)H_{\ast}(HGC). However, although embedding calculus allows hairy graphs with any number of loops, configuration space integral was successful only for BCR graphs with zero or one loop. This caused much less information for π∗​(𝒦¯n,j,ℚ)\pi_{\ast}(\overline{\mathcal{K}}_{n,j},\mathbb{Q}) or H∗​(𝒦¯n,j,ℚ)H_{\ast}(\overline{\mathcal{K}}_{n,j},\mathbb{Q}) given by configuration space integral.

In the author’s previous paper [Yos], we have succeeded in conducting configuration space integral for some 2-loop BCR graphs, and in giving a new non-trivial (co)cycle of the space of long embeddings (including n−j=2n-j=2). However, we haven’t been able to give more examples, due to much more complicated combinatorics of H∗​(G​C)H_{\ast}(GC) than H∗​(H​G​C)H_{\ast}(HGC).

In this paper, we relate the two graph homologies H∗​(G​Cn,j)H_{\ast}(GC^{n,j}) and H∗​(H​G​Cn,j)H_{\ast}(HGC^{n,j}) at the top (non-degenerate) part. We write the top part of H∗​(H​G​C)H_{\ast}(HGC) as ℬ\mathcal{B} and write the top part of H∗​(G​C)H_{\ast}(GC) as 𝒜B​C​R\mathcal{A}_{BCR}, though 𝒜B​C​R\mathcal{A}_{BCR} is quite modified as explained later. We focus on the case n−jn-j is even because the author is most interested in the case n−j=2n-j=2. The following is our main result.

Theorem 0.1.

There is a monomorphism χ∗\chi_{\ast} from ℬ\mathcal{B} to 𝒜B​C​R\mathcal{A}_{BCR}.

The proof of our main result is very similar to the proof by Bar-Natan[Bar] of PBW (Poincaré–Birkhoff–Witt) isomorphisms between ℬ\mathcal{B}, the space of open Jacobi diagrams, and 𝒜​(S1)\mathcal{A}(S_{1}), the space of Jacobi diagrams on the unit circle. As Bar-Natan showed another isomorphism between 𝒜​(S1)\mathcal{A}(S_{1}) and the space of chord diagrams 𝒜c​(S1)\mathcal{A}^{c}(S_{1}), we have the following. Let 𝒜B​C​Rc\mathcal{A}^{c}_{BCR} be the space of BCR chord diagrams defined in Definition 1.17.

Theorem 0.2.

There is an isomorphism ι∗\iota_{\ast} from 𝒜B​C​Rc\mathcal{A}^{c}_{BCR} to 𝒜B​C​R\mathcal{A}_{BCR}.

It might be possible to apply our main result to give new nontrivial (co)cycles of 𝒦¯n,j\overline{\mathcal{K}}_{n,j} by the following steps: First, compute some part (two-loop part, for example) of ℬ\mathcal{B} by relating it with (anti-)symmetric polynomials as in [CCTW]. Next, by Theorem 0.1, we obtain some nontrivial subspace of 𝒜B​C​R\mathcal{A}_{BCR}. This gives some non-trivial graph cocycle H=∑iai​ΓiH=\sum_{i}a_{i}\Gamma_{i}. (See the next paragraph.) So we can perform configuration space integral

I¯​(H)=∑iai​∫ConfΩ¯​(Γi)\overline{I}(H)=\sum_{i}a_{i}\int_{\text{Conf}}\overline{\Omega}(\Gamma_{i})

associated with HH (with the correction term c¯\overline{c} in [Yos]). Finally, conduct pairing argument in [Yos] between the cocycle I¯​(H)\overline{I}(H) and a suitably-constructed cycle of 𝒦¯n,j\overline{\mathcal{K}}_{n,j}. Then I¯​(H)\overline{I}(H) might detect the non-triviality of this cycle.

However, it should be emphasized that the (top) graph homology 𝒜B​C​R\mathcal{A}_{BCR} is modified in this paper: 𝒜B​C​R\mathcal{A}_{BCR} has no relation corresponding to contractions of chords. (See Definition 1.12.) Hence, a graph cocycle HH constructed through a weight system ww on 𝒜B​C​R\mathcal{A}_{BCR}

H=∑Γw​(Γ)Aut​(Γ)​ΓH=\sum_{\Gamma}\frac{w(\Gamma)}{\text{Aut}(\Gamma)}\Gamma

may not be a genuine graph cocycle used in [Yos], because contractions of chords may not be canceled. One possible solution will be adding correction terms using the data of paths of immersions that 𝒦¯n,j\overline{\mathcal{K}}_{n,j} has. We deal with the correction terms in the author’s future work in preparation [Yos2].

The second result holds even if we impose relations on chords to the spaces 𝒜B​C​Rc\mathcal{A}^{c}_{BCR} to 𝒜B​C​R\mathcal{A}_{BCR}. Let 𝒜¯B​C​Rc\overline{\mathcal{A}}^{c}_{BCR} to 𝒜¯B​C​R\overline{\mathcal{A}}_{BCR} be the quotient space.

Theorem 0.3.

There is an isomorphism ι¯\overline{\iota} from 𝒜¯B​C​Rc\overline{\mathcal{A}}^{c}_{BCR} to 𝒜¯B​C​R\overline{\mathcal{A}}_{BCR}.

At present, the author does not know there is a monomorphism from ℬ\mathcal{B} to the genuine graph homology 𝒜¯B​C​R\overline{\mathcal{A}}_{BCR}. But it holds for all the cases the author can compute 𝒜¯B​C​R\overline{\mathcal{A}}_{BCR}, including the simplest 2-loop case in [Yos]. Note that the map is not necessarily surjective even in the case the target is the quotient space 𝒜¯B​C​R\overline{\mathcal{A}}_{BCR}.

Question 0.4.

Is the map χ¯∗\overline{\chi}_{\ast} from ℬ\mathcal{B} to 𝒜¯B​C​R\overline{\mathcal{A}}_{BCR} injective?

As a final remark, we go back to Arone and Turchin’s paper [AT2]. In [AT2], the graph homology H∗​(G​C)H_{\ast}(GC), which is written there as H∗​(ℰπj,n)H_{\ast}(\mathcal{E}^{j,n}_{\pi}), is constructed from the homology of a derived mapping space between Ω\Omega modules:

H∗(RmodΩh(H~∗(Sm⁣∙),H¯∗(C∙(ℝn)))H_{\ast}\left(\underset{\Omega}{\text{Rmod}}^{h}(\widetilde{H}_{\ast}(S^{m\bullet}),\overline{H}_{\ast}(C_{\bullet}(\mathbb{R}^{n}))\right)

by an injective resolution of the target. In [AT2], another graph homology H∗​(𝒦πj,n)H_{\ast}(\mathcal{K}^{j,n}_{\pi}) is introduced by a projective resolution of the source. By taking both projective and injective resolutions, one obtains one more graph homology H∗​(𝒴πj,n)H_{\ast}(\mathcal{Y}^{j,n}_{\pi}), which does not explicitly appear in [AT2]. Since the origin is the same, the three graph homologies are isomorphic. It would be interesting to see directly the differences between H∗​(𝒴πj,n)H_{\ast}(\mathcal{Y}^{j,n}_{\pi}) and H∗​(G​C)H_{\ast}(GC); the duals of H∗​(G​C)H_{\ast}(GC) and H∗​(𝒴πj,n)H_{\ast}(\mathcal{Y}^{j,n}_{\pi}) have almost the same graphs. A similar question arises for differences between 𝒜B​C​Rc\mathcal{A}^{c}_{BCR} and the top part of H∗​(𝒦πj,n)H_{\ast}(\mathcal{K}^{j,n}_{\pi}). See Remark 1.21.

This paper is organized as follows. In Section 1, we define three spaces of graphs: ℬ\mathcal{B}, 𝒜B​C​R\mathcal{A}_{BCR} and 𝒜B​C​Rc\mathcal{A}^{c}_{BCR}. Our main result is stated in Section 2. After introducing several notations for proof in Section 3, we show our main result in Section 4.

Acknowledgement

The author is deeply grateful to Victor Turchin for explaining to the author a lot about their work on graph homology. The author would like to thank Tadayuki Watanabe for motivating and inspiring discussion with the author. This paper was written during the author’s three-month stay at Kansas State University. The author appreciates their hospitality. This research was supported by Forefront Physics and Mathematics Program to Drive Transformation (FoPM), a World-leading Innovative Graduate Study (WINGS) Program, The University of Tokyo.

1 Definition of graph homologies

1.1 The modified BCR graph homology 𝒜B​C​R\mathcal{A}_{BCR}

Here, we define the spaces 𝒜B​C​R\mathcal{A}_{BCR} and 𝒜¯B​C​R\overline{\mathcal{A}}_{BCR} of BCR graphs. This space is almost equivalent to the dual of the top graph cohomology H0​(G​C)H^{0}(GC) defined in [Yos] as H0​(𝒟)H^{0}(\mathcal{D}). However, we add an extra condition (the last condition in the following definition) to the graphs. Moreover, we exclude some relations on chords.

Definition 1.1 (BCR graphs).

A BCR graph is a connected graph that satisfies the following.

  • •

    There are (at most) two types of vertices: external vertices drawn in black and internal vertices drawn in white. The graph must have at least one external vertex.

  • •

    There are (at most) two types of edges: dashed edges and solid edges.

  • •

    Each internal vertex has exactly three dashed edges and no solid edge.

  • •

    Each external vertex has exactly one dashed edge and at most two solid edges.

  • •

    The restriction to solid edges of the graph forms a disjoint sum of broken lines such as . In particular, neither double solid edges nor small solid loops are allowed.

Remark 1.2.

In Definition 1.1, small dashed loops , double dashed edges and multiple edges are allowed. However, some of them vanish by relations we later define.

Definition 1.3.

We define two ways of coloring BCR graphs: (I) odd case and (II) even case.

  • (I)

    We orient each edge and label the vertices from 11 to nn if the number of vertices is nn.

  • (II)

    We number the edges from (1)(1) to (m)(m) if the number of edges is mm.

We say two colorings of the isomorphic underlying graph are the same (resp. opposite) orientation if they are transferred from each other by even (resp. odd) permutation.

Definition 1.4.

We write 𝒟\mathcal{D} for the vector space over ℚ\mathbb{Q} generated by colored BCR graphs.

Definition 1.5.

We write 𝒜B​C​R\mathcal{A}_{BCR} for the quotient vector space of 𝒟\mathcal{D} by IHX, STU, and orientation relations. We define STU and IHX relations below.

Definition 1.6.

We define IHX relations of colored BCR graphs as in Figure 1 below. In Figure 1, graphs D1,D2,D3D_{1},D_{2},D_{3} are allowed to have any colors so that contractions of the middle edges give three isomorphic graphs with the same orientation (after transposition to the left-hand side). See [Yos] for the sign rule for contraction of edges.

D1D_{1}

= D2D_{2} - D3D_{3}.

Figure 1: IHX relation
Remark 1.7.

As a consequence of an IHX relation, a small dashed loop at an internal vertex that is connected to another internal vertex vanishes:

=0.\leavevmode\hbox to81pt{\vbox to64pt{\pgfpicture\makeatletter\hbox{\hskip 40.5pt\lower-23.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{{}}{{}}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{ } {}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{5.0pt,4.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{-40.0pt}{40.0pt}{40.0pt}{40.0pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{-2.0pt}\pgfsys@moveto{3.0pt}{-2.0pt}\pgfsys@curveto{3.0pt}{-0.34312pt}{1.65688pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-1.65688pt}{1.0pt}{-3.0pt}{-0.34312pt}{-3.0pt}{-2.0pt}\pgfsys@curveto{-3.0pt}{-3.65688pt}{-1.65688pt}{-5.0pt}{0.0pt}{-5.0pt}\pgfsys@curveto{1.65688pt}{-5.0pt}{3.0pt}{-3.65688pt}{3.0pt}{-2.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-2.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{-20.0pt}\pgfsys@moveto{3.0pt}{-20.0pt}\pgfsys@curveto{3.0pt}{-18.34312pt}{1.65688pt}{-17.0pt}{0.0pt}{-17.0pt}\pgfsys@curveto{-1.65688pt}{-17.0pt}{-3.0pt}{-18.34312pt}{-3.0pt}{-20.0pt}\pgfsys@curveto{-3.0pt}{-21.65688pt}{-1.65688pt}{-23.0pt}{0.0pt}{-23.0pt}\pgfsys@curveto{1.65688pt}{-23.0pt}{3.0pt}{-21.65688pt}{3.0pt}{-20.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-20.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{5.0pt,4.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{-5.0pt}\pgfsys@lineto{0.0pt}{-17.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=0.
Definition 1.8.

We define STU relations (I)(II) of colored BCR graphs as in Figure 2 below. In Figure 2, graphs DiD_{i} are allowed to have any colors so that contractions of the middle edges (circled in red) give isomorphic graphs with the same orientation. In the case (I), the bottom vertex of the graph D1D_{1} has one or two solid edges. In the case (II), the bottom vertex of the graph D1D_{1} has no solid edge.

Refer to caption
Figure 2: STU relations (I)(II)
Remark 1.9.

As a consequence of an STU relation in the odd case (Definition 1.3), a small dashed loop at an internal vertex that is connected to an external vertex vanishes:

=0(odd case).\leavevmode\hbox to81pt{\vbox to64pt{\pgfpicture\makeatletter\hbox{\hskip 40.5pt\lower-23.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{{}}{{}}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{ } {}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{5.0pt,4.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{-40.0pt}{40.0pt}{40.0pt}{40.0pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{-2.0pt}\pgfsys@moveto{3.0pt}{-2.0pt}\pgfsys@curveto{3.0pt}{-0.34312pt}{1.65688pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-1.65688pt}{1.0pt}{-3.0pt}{-0.34312pt}{-3.0pt}{-2.0pt}\pgfsys@curveto{-3.0pt}{-3.65688pt}{-1.65688pt}{-5.0pt}{0.0pt}{-5.0pt}\pgfsys@curveto{1.65688pt}{-5.0pt}{3.0pt}{-3.65688pt}{3.0pt}{-2.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-2.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{-20.0pt}\pgfsys@moveto{3.0pt}{-20.0pt}\pgfsys@curveto{3.0pt}{-18.34312pt}{1.65688pt}{-17.0pt}{0.0pt}{-17.0pt}\pgfsys@curveto{-1.65688pt}{-17.0pt}{-3.0pt}{-18.34312pt}{-3.0pt}{-20.0pt}\pgfsys@curveto{-3.0pt}{-21.65688pt}{-1.65688pt}{-23.0pt}{0.0pt}{-23.0pt}\pgfsys@curveto{1.65688pt}{-23.0pt}{3.0pt}{-21.65688pt}{3.0pt}{-20.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-20.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{5.0pt,4.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{-5.0pt}\pgfsys@lineto{0.0pt}{-17.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=0\quad(\text{odd case}).
Remark 1.10.

As a consequence of an orientation relation in the even case, a double dashed edge vanishes.

=0(even case).\leavevmode\hbox to113pt{\vbox to21pt{\pgfpicture\makeatletter\hbox{\hskip 5.5pt\lower-10.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{{}}{{}}\pgfsys@setlinewidth{1.0pt}\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-2.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{1.65688pt}{-0.34312pt}{3.0pt}{-2.0pt}{3.0pt}\pgfsys@curveto{-3.65688pt}{3.0pt}{-5.0pt}{1.65688pt}{-5.0pt}{0.0pt}\pgfsys@curveto{-5.0pt}{-1.65688pt}{-3.65688pt}{-3.0pt}{-2.0pt}{-3.0pt}\pgfsys@curveto{-0.34312pt}{-3.0pt}{1.0pt}{-1.65688pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-2.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{104.0pt}{0.0pt}\pgfsys@moveto{107.0pt}{0.0pt}\pgfsys@curveto{107.0pt}{1.65688pt}{105.65688pt}{3.0pt}{104.0pt}{3.0pt}\pgfsys@curveto{102.34312pt}{3.0pt}{101.0pt}{1.65688pt}{101.0pt}{0.0pt}\pgfsys@curveto{101.0pt}{-1.65688pt}{102.34312pt}{-3.0pt}{104.0pt}{-3.0pt}\pgfsys@curveto{105.65688pt}{-3.0pt}{107.0pt}{-1.65688pt}{107.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{104.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{5.0pt,4.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{40.0pt}{10.0pt}{60.0pt}{10.0pt}{100.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{5.0pt,4.0pt}{0.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{40.0pt}{-10.0pt}{60.0pt}{-10.0pt}{100.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =0\quad(\text{even case}).
Definition 1.11.

We write 𝒜¯B​C​R\overline{\mathcal{A}}_{BCR} for the quotient vector space of 𝒟\mathcal{D} by IHX, STU, orientation relations and chord relations. We define chord relations below.

Definition 1.12.

We define chord relations (I)(II) of colored BCR graphs as in Figure 3 below. In Figure 3, graphs DiD_{i} are allowed to have any colors so that contractions of the middle edges give isomorphic graphs with the same orientation. Here, We only use chord relations in which all the graphs DiD_{i} are modified BCR graphs. Namely, if the contracted graph has a loop consisting of only solid edges and external vertices, we do not consider the relation as a chord relation. In the case (I), the bottom vertex of the graph D1D_{1} has one or two solid edges. In the case (II), the bottom vertex of the graph D1D_{1} has no solid edge.

Refer to caption
Figure 3: Chord relations (I)(II)
Remark 1.13.

We exclude contractions of chords that yield loops consisting of only solid edges for the following reason: if the dimension jj of the space of embedding 𝒦n,j\mathcal{K}_{n,j} is large enough comparing the order (size) kk, more precisely j≥2​kj\geq 2k, the integral by such graphs vanish by dimensional reason. This argument is introduced in Section 8 of [Yos]. Our second result 0.2 holds even if we admit graphs with solid loops and add chord relations that include such graphs.

1.2 The hairy graph homology ℬ\mathcal{B}

We define the space ℬ\mathcal{B} of hairy graphs. The space ℬ\mathcal{B} is an analog of the space of open Jacobi diagrams modulo IHX and AS relations (and they coincide in the odd case).

Definition 1.14 (Hairy graphs).

A hairy graph is a connected graph that satisfies the following.

  • •

    There are (at most) two types of vertices: external vertices drawn in black and internal vertices drawn in white. The graph must have at least one external vertex.

  • •

    There is only one type of edges, dashed.

  • •

    Each external vertex has exactly one dashed edge. Each internal vertex has exactly three dashed edges.

We can define color and orientation of hairy graphs by those of BCR graphs.

Definition 1.15.

We write 𝒞\mathcal{C} for the vector space over ℚ\mathbb{Q} generated by colored hairy graphs. We define the space ℬ\mathcal{B} as the quotient space of 𝒞\mathcal{C} by orientation relations and IHX relations. In the odd case, we add the following relation (See Remark 1.9).

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Remark 1.16.

In [AT2], Arone and Turchin introduced a graph complex ℰπj,n\mathcal{E}^{j,n}_{\pi}, which is shown to be isomorphic to the rational homotopy group π∗​(𝒦n,j)⊗ℚ\pi_{\ast}(\mathcal{K}_{n,j})\otimes\mathbb{Q} for n−j≥3n-j\geq 3 by Fresse, Turchin and Willwacher [FTW]. The top part of H∗​(ℰπj,n)H_{\ast}(\mathcal{E}^{j,n}_{\pi}) is isomorphic to ℬ\mathcal{B}. In the odd case, ℬ\mathcal{B} is isomorphic to the space of hairy graphs modulo IHX and AS relations [Bar].

1.3 The space of BCR chord diagrams 𝒜B​C​Rc\mathcal{A}^{c}_{BCR}

Next, we define the spaces of BCR chord diagrams 𝒜B​C​Rc\mathcal{A}^{c}_{BCR} and 𝒜¯B​C​Rc\overline{\mathcal{A}}^{c}_{BCR}. These spaces are analogs of the space of chord diagrams 𝒜c​(S1)\mathcal{A}^{c}(S^{1}), which appears in the theory of Vassiliev knot invariants.

Definition 1.17 (BCR chord diagrams).

A BCR chord diagram is a connected graph that satisfies the following.

  • •

    There is only one type of vertices: external vertices drawn in black.

  • •

    There are two types of edges, solid and dashed.

  • •

    Each external vertex has exactly one dashed edge and at most two solid edges.

  • •

    The restriction to solid edges of the graph forms a disjoint union of broken lines. See Definition 1.1.

We can define color and orientation of BCR chord diagrams by those of BCR graphs.

Definition 1.18.

We write 𝒟c\mathcal{D}^{c} for the vector space over ℚ\mathbb{Q} generated by colored BCR chord diagrams. We define 𝒜B​C​Rc\mathcal{A}^{c}_{BCR} as the quotient space of 𝒟c\mathcal{D}^{c} by orientation relations and 4​T4T relations, which are defined below.

Definition 1.19.

We define 4​T4T relations (I)(II)(III) of colored BCR chord diagrams as in Figure 4 below. In Figure 4, graphs DiD_{i} are allowed to have any colors so that S=D1−D2S=D_{1}-D_{2} and S=D3−D4S=D_{3}-D_{4} are both STU relations for some colored BCR graph SS. The graph SS is uniquely determined after orientation relations in 𝒜B​C​R\mathcal{A}_{BCR}.

Refer to caption
Figure 4: 4T relations (I)(II)(III)
Definition 1.20.

We define 𝒜¯B​C​Rc\overline{\mathcal{A}}^{c}_{BCR} as the quotient space of 𝒟c\mathcal{D}^{c} by orientation relations, 4​T4T relations and chord relations. Here, we only use chord relations in which all the graphs DiD_{i} are BCR chord diagrams.

Remark 1.21.

In [AT2], Arone and Turchin introduced a complex 𝒦πj,n\mathcal{K}^{j,n}_{\pi} that is quasi-isomorphic to ℰπj,n\mathcal{E}^{j,n}_{\pi}. Though 𝒦πj,n\mathcal{K}^{j,n}_{\pi} is a little difficult to describe, its dual complex H​Hπj,nHH^{j,n}_{\pi} is described in terms of graphs [AT2]. The top (non-degenerate) part of H​Hπj,nHH^{j,n}_{\pi} is generated by BCR chord diagrams. However, they allow not only broken lines but also trees as the restriction to solid edges. They also impose Arnold relations that come from relations in H∗​(Ck​(ℝm))H^{\ast}(C_{k}(\mathbb{R}^{m})) and H∗​(Ck​(ℝn))H^{\ast}(C_{k}(\mathbb{R}^{n})). As a consequence of Arnold relations (with respect to solid edges), trees of solid edges can be transformed into a linear combination of broken lines. Our 4​T4T relations correspond to the differential dd of H​Hπj,nHH^{j,n}_{\pi} (with Arnold relations with respect to dashed edges). So far, the author finds nothing related to chord relations in the complex H​Hj,nπHH_{j,n}^{\pi}.

2 Main Result

2.1 The map χ∗:ℬ→𝒜B​C​R\chi_{\ast}:\mathcal{B}\rightarrow\mathcal{A}_{BCR}

Proposition 2.1.

We define the map χ:𝒞→𝒟\chi:\mathcal{C}\rightarrow\mathcal{D} by the inclusion. Then this map induces χ∗:ℬ→𝒜B​C​R\chi_{\ast}:\mathcal{B}\rightarrow\mathcal{A}_{BCR}.

Theorem 2.2.

The map χ∗:ℬ→𝒜B​C​R\chi_{\ast}:\mathcal{B}\rightarrow\mathcal{A}_{BCR} is a monomorphism.

2.2 The map ι∗:𝒜c→𝒜B​C​R\iota_{\ast}:\mathcal{A}^{c}\rightarrow\mathcal{A}_{BCR}

Proposition 2.3.

We define the map ι:𝒟c→𝒟\iota:\mathcal{D}^{c}\rightarrow\mathcal{D} by the inclusion. Then this map induces the maps ι∗:𝒜B​C​Rc→𝒜B​C​R\iota_{\ast}:\mathcal{A}^{c}_{BCR}\rightarrow\mathcal{A}_{BCR} and ι¯∗:𝒜¯B​C​Rc→𝒜¯B​C​R\overline{\iota}_{\ast}:\overline{\mathcal{A}}^{c}_{BCR}\rightarrow\overline{\mathcal{A}}_{BCR}.

Theorem 2.4.

Both of the maps ι∗:𝒜B​C​Rc→𝒜B​C​R\iota_{\ast}:\mathcal{A}^{c}_{BCR}\rightarrow\mathcal{A}_{BCR} and ι¯∗:𝒜¯B​C​Rc→𝒜¯B​C​R\overline{\iota}_{\ast}:\overline{\mathcal{A}}^{c}_{BCR}\rightarrow\overline{\mathcal{A}}_{BCR} are isomorphisms.

3 Preliminary for proof of Main Result

Here, we introduce notations used in Section 4.

Definition 3.1.

A solid component of a BCR graph DD is a component of solid broken lines. That is, a solid component consists of black vertices v1,v2,…,vnv_{1},v_{2},\dots,v_{n} and solid edges ei=(vi,vi+1)e_{i}=(v_{i},v_{i+1}) for any i=1,…,n−1i=1,\dots,n-1, where v1,v2,…,vnv_{1},v_{2},\dots,v_{n} have no other solid edges. The solid type T​(D)T(D) of DD is the collection (in descending order) of lengths of solid components. For example, the graph in Figure 5 has the solid type (2,2)(2,2).

Refer to caption
Figure 5: A graph with the solid type (2,2)
Notation 3.2.

Let 𝒞k\mathcal{C}_{k} be the subspace of 𝒞\mathcal{C} spanned by colored hairy graphs with at most kk external vertices. Let 𝒟k\mathcal{D}_{k} be the subspace of 𝒟\mathcal{D} spanned by colored BCR graphs with at most kk external vertices. Clearly χ:𝒞→𝒟\chi:\mathcal{C}\rightarrow\mathcal{D} restricts to χ:𝒞k→𝒟k\chi:\mathcal{C}_{k}\rightarrow\mathcal{D}_{k}. Let ℐk\mathcal{I}_{k} be the subspace of 𝒟k\mathcal{D}_{k} spanned by STU relations of graphs with at most kk external vertices.

Notation 3.3.

Let DD be a colored BCR graph with kk external vertices. Assume that solid components of DD are in a row as in Figure 6. For π∈𝔖k\pi\in\mathfrak{S}_{k}, let (π​D)u(\pi D)_{u} be the colored graph obtained from DD by permutating dashed edges so that iith edge from the left goes to the σ​(i)\sigma(i)th. Then we set

π​D=sign​(π)​(π​D)u∈𝒟.\pi D=\text{sign}(\pi)(\pi D)_{u}\in\mathcal{D}.

Let 𝔖k​(D)\mathfrak{S}_{k}(D) be the subset of 𝔖k\mathfrak{S}_{k} which consists of permutations that preserve solid components. That is, if π∈𝔖k\pi\in\mathfrak{S}_{k}, the iith and π​(i)\pi(i)th edges must be connected to the same solid component. If the solid type of DD is T​(D)=(i1,…,is)T(D)=(i_{1},\dots,i_{s}), 𝔖k​(D)\mathfrak{S}_{k}(D) is isomorphic to

∏l=1s𝔖il.\prod_{l=1}^{s}\mathfrak{S}_{i_{l}}.
Refer to caption
Figure 6: A permutation of DD
Notation 3.4.

Let DD be a colored BCR graph with kk external vertices. Let UiU_{i} be a transposition in 𝔖k​(D)\mathfrak{S}_{k}(D). We define Si​DS_{i}D as a colored BCR graph with k−1k-1 vertices so that the equation

Si​D=D−Ui​DS_{i}D=D-U_{i}D

is an STU relation. The graph Si​DS_{i}D is determined uniquely after orientation relation in 𝒟\mathcal{D}. See Figure 7.

Refer to caption
Figure 7: Example of Si​DS_{i}D

4 Proof of Main Result

The proof of Proposition 2.1 and 2.3 is trivial by the definitions of their relations. We prove Theorem 2.2 and Theorem 2.4 in the following. The proof is very similar to the proof of [Bar, Theorem 8] and [Bar, Theorem 6].

4.1 χ∗:ℬ→𝒜B​C​R\chi_{\ast}:\mathcal{B}\rightarrow\mathcal{A}_{BCR} is a monomorphism

Proof of Theorem 2.2.

We construct the left inverse σ∗\sigma_{\ast} of the map χ∗\chi_{\ast}. We construct inductively the map σk:𝒟k→ℬ\sigma_{k}:\mathcal{D}_{k}\rightarrow\mathcal{B} which satisfies the following.

  • (Σ​1\Sigma 1)

    σk∘χk:𝒞k→ℬ\sigma_{k}\circ\chi_{k}:\mathcal{C}_{k}\rightarrow\mathcal{B} is the projection.

  • (Σ​2\Sigma 2)

    σk\sigma_{k}(orientation relation) =0.=0.

  • (Σ​3\Sigma 3)

    σk\sigma_{k}(IHX relation) =0.=0.

  • (Σ​4\Sigma 4)

    σk​(ℐk)=0.\sigma_{k}(\mathcal{I}_{k})=0.

First, we define σ1:𝒟1→ℬ\sigma_{1}:\mathcal{D}_{1}\rightarrow\mathcal{B} as the identity. Clearly σ1\sigma_{1} satisfies (Σ​1\Sigma 1)(Σ​2\Sigma 2)(Σ​3\Sigma 3)(Σ​4\Sigma 4). Next we assume that σk−1:𝒟k−1→ℬ\sigma_{k-1}:\mathcal{D}_{k-1}\rightarrow\mathcal{B} is constructed so that it satisfies (Σ​1\Sigma 1)(Σ​2\Sigma 2)(Σ​3\Sigma 3)(Σ​4\Sigma 4). Then we define σk:𝒟k→ℬ\sigma_{k}:\mathcal{D}_{k}\rightarrow\mathcal{B} by

σk​(D)=1|𝔖k​(D)|​∑π∈𝔖k​(D)σk−1​(ΓD​(π))\sigma_{k}(D)=\frac{1}{|\mathfrak{S}_{k}(D)|}\sum_{\pi\in\mathfrak{S}_{k}(D)}\sigma_{k-1}(\Gamma_{D}(\pi))

for a graph DD with kk external vertices.111Remember that Bar-Natan’s original construction was σk​(D)=1k!​(Dc​c+∑π∈​𝔖kΛD​(π))\sigma_{k}(D)=\frac{1}{k!}(D^{cc}+\sum_{\pi_{\in}\mathfrak{S}_{k}}\Lambda_{D}(\pi)). We do not use Dc​cD^{cc} part. Here ΓD​(π)\Gamma_{D}(\pi) is an element of 𝒟k−1\mathcal{D}_{k-1} determined uniquely after orientation relations. In 𝒜B​C​R\mathcal{A}_{BCR}, the element ΓD​(π)\Gamma_{D}(\pi) satisfies

D−π​D=ΓD​(π).D-\pi D=\Gamma_{D}(\pi).

We explain the construction of ΓD​(π)\Gamma_{D}(\pi). If π\pi is the elementary transposition Ui=(i,i+1)∈𝔖k​(D)U_{i}=(i,i+1)\in\mathfrak{S}_{k}(D), we set ΓD​(π=Ui)=Si​D\Gamma_{D}(\pi=U_{i})=S_{i}D. For a general permutation σ∈𝔖k​(D)\sigma\in\mathfrak{S}_{k}(D), write σ\sigma as the product of elementary transpositions of 𝔖k​(D)\mathfrak{S}_{k}(D): π=Ui1​…​Uin\pi=U_{i_{1}}\dots U_{i_{n}}. Then

ΓD​(π)=∑l=1nSil​Uil+1​…​Uin​Γ.\Gamma_{D}(\pi)=\sum_{l=1}^{n}S_{i_{l}}U_{i_{l+1}}\dots U_{i_{n}}\Gamma.
Example 4.1.

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Then we have

ΓD​(π)={Ewhen ​π=(1 2),(2 3),(1 2 3),(1 3 2)∈𝔖3,0when ​π=i​d,(1 3)∈𝔖3.\Gamma_{D}(\pi)=\begin{cases}E&\text{when\ }\pi=(1\ 2),(2\ 3),(1\ 2\ 3),(1\ 3\ 2)\in\mathfrak{S}_{3},\\ 0&\text{when\ }\pi=id,(1\ 3)\in\mathfrak{S}_{3}.\\ \end{cases}

So we have

σ4​(D)\displaystyle\sigma_{4}(D) =16​∑π∈𝔖3σ3​(ΓD​(π))=23​σ3​(E)=23​σ2​(F)=23​F.\displaystyle=\frac{1}{6}\sum_{\pi\in\mathfrak{S}_{3}}\sigma_{3}(\Gamma_{D}(\pi))=\frac{2}{3}\sigma_{3}(E)=\frac{2}{3}\sigma_{2}(F)=\frac{2}{3}F.

Similarly, we have

σ4​(D′)=13​σ3​(E)=13​σ2​(F)=13​F\sigma_{4}(D^{\prime})=\frac{1}{3}\sigma_{3}(E)=\frac{1}{3}\sigma_{2}(F)=\frac{1}{3}F

Note that F=E=D+D′F=E=D+D^{\prime} are STU relations.

We must show that σk​(D)\sigma_{k}(D) does not depend on choices of the expression π=Ui1​…​Uin\pi=U_{i_{1}}\dots U_{i_{n}}. Let ℳk​(D)\mathcal{M}_{k}(D) be the monoid generated by elementary transpositions of 𝔖k​(D)\mathfrak{S}_{k}(D). We have constructed the map

ΛD​(π)=σk−1∘ΓD:ℚ​[ℳk​(D)]→ℬ\Lambda_{D}(\pi)=\sigma_{k-1}\circ\Gamma_{D}:\mathbb{Q}[\mathcal{M}_{k}(D)]\rightarrow\mathcal{B}

from the monoid ring of ℳk​(D)\mathcal{M}_{k}(D). To show it defines a map from ℚ​[𝔖k​(D)]\mathbb{Q}[\mathfrak{S}_{k}(D)], it is enough to show the kernel of ΛD\Lambda_{D} includes the kernel 𝒦\mathcal{K} of the map ℚ​[ℳk​(D)]→ℚ​[𝔖k​(D)]\mathbb{Q}[\mathcal{M}_{k}(D)]\rightarrow\mathbb{Q}[\mathfrak{S}_{k}(D)]. The kernel 𝒦\mathcal{K} is the both-sided ideal generated by

  • (G1)

    Ui2−i​dU_{i}^{2}-id,

  • (G2)

    Ui​Uj−Uj​Ui=i​d​(|i−j|>1)U_{i}U_{j}-U_{j}U_{i}=id\ (|i-j|>1),

  • (G3)

    Ui​Ui−1​Ui−Ui−1​Ui​Ui−1U_{i}U_{i-1}U_{i}-U_{i-1}U_{i}U_{i-1}.

Here the UiU_{i} s in each relation must be elements of the same 𝔖il\mathfrak{S}_{i_{l}} of 𝔖k​(D)=∏l=1s𝔖il\mathfrak{S}_{k}(D)=\prod_{l=1}^{s}\mathfrak{S}_{i_{l}}.

Let DD be a colored graph of the even case

(k)(k)(l)(l)(m)(m)

,

where the (i−1)(i-1)th, ii-th, (i+1)(i+1)-th, jj-th, (j+1)(j+1)-th dashed edges are drawn. Without loss of genericity, we assume that k<l<mk<l<m. We show that ΛD\Lambda_{D} vanishes on (G1), (G2) and (G3). A completely similar proof works for the odd case.

(G1)

ΛD​(Ui2)\displaystyle\Lambda_{D}(U_{i}^{2})
=\displaystyle= σk−1​(Si​D+Si​Ui​D)\displaystyle\sigma_{k-1}(S_{i}D+S_{i}U_{i}D)
=\displaystyle= σk−1​(Si​D−Si​D)\displaystyle\sigma_{k-1}(S_{i}D-S_{i}D)
=\displaystyle= 0.\displaystyle 0.

(G2)

ΛD​(Ui​Uj−Uj​Ui)\displaystyle\Lambda_{D}(U_{i}U_{j}-U_{j}U_{i})
=\displaystyle= σk−1​(Sj​D+Si​Uj​D−Si​D−Sj​Ui​D)\displaystyle\sigma_{k-1}(S_{j}D+S_{i}U_{j}D-S_{i}D-S_{j}U_{i}D)
=\displaystyle= σk−1​(Sj​D−Sj​Ui​D−Si​D+Si​Uj​D).\displaystyle\sigma_{k-1}(S_{j}D-S_{j}U_{i}D-S_{i}D+S_{i}U_{j}D).

By STU relation in ℐk−1\mathcal{I}_{k-1},

Sj​D−Sj​Ui​D=Ds=Si​D−Si​Uj​DS_{j}D-S_{j}U_{i}D=D_{s}=S_{i}D-S_{i}U_{j}D

for the graph DsD_{s} in Figure 8, which has k−2k-2 external vertices. Since σk−1\sigma_{k-1} satisfies (Σ​4\Sigma 4) by the induction hypothesis, we have ΛD​(Ui​Uj−Uj​Ui)=0\Lambda_{D}(U_{i}U_{j}-U_{j}U_{i})=0.

Refer to caption
Figure 8: Graph DsD_{s} in (G2) (even case)

(G3)

ΛD​(Ui​Ui−1​Ui−Ui−1​Ui​Ui−1)\displaystyle\Lambda_{D}(U_{i}U_{i-1}U_{i}-U_{i-1}U_{i}U_{i-1})
=\displaystyle= σk−1​(Si​D+Si−1​Ui​D+Si​Ui−1​Ui​D−Si−1​D−Si​Ui−1​D−Si−1​Ui​Ui−1​D)\displaystyle\sigma_{k-1}(S_{i}D+S_{i-1}U_{i}D+S_{i}U_{i-1}U_{i}D-S_{i-1}D-S_{i}U_{i-1}D-S_{i-1}U_{i}U_{i-1}D)
=\displaystyle= σk−1​(Si​D−Si−1​Ui​Ui−1​D+Si−1​Ui​D−Si​Ui−1​D+Si​Ui−1​Ui​D−Si−1​D)\displaystyle\sigma_{k-1}(S_{i}D-S_{i-1}U_{i}U_{i-1}D+S_{i-1}U_{i}D-S_{i}U_{i-1}D+S_{i}U_{i-1}U_{i}D-S_{i-1}D)
=\displaystyle= σk−1​(DI+DH+DX).\displaystyle\sigma_{k-1}(D_{I}+D_{H}+D_{X}).

Where DID_{I}, DHD_{H} and DXD_{X} are graphs in Figure 9 and DI+DH+DXD_{I}+D_{H}+D_{X} satisfies an IHX relation. We used STU relation in ℐk−1\mathcal{I}_{k-1} in the last equation, Since σk−1\sigma_{k-1} satisfies (Σ​3\Sigma 3) by the induction hypothesis, we have ΛD​(Ui​Ui−1​Ui−Ui−1​Ui​Ui−1)=0\Lambda_{D}(U_{i}U_{i-1}U_{i}-U_{i-1}U_{i}U_{i-1})=0.

Refer to caption
Figure 9: Graphs DI,DH,DXD_{I},D_{H},D_{X} in (G3) (even case)

Next we prove that σk\sigma_{k} satisfies (Σ​1\Sigma 1) (Σ​2\Sigma 2) (Σ​3\Sigma 3) (Σ​4\Sigma 4).

(Σ​1\Sigma 1) For a hairy graph CC,

σk∘χk​(C)=σk​(C)=C.\displaystyle\sigma_{k}\circ\chi_{k}(C)=\sigma_{k}(C)=C.

(Σ​2\Sigma 2) If the orientation of DD is reversed (resp. preserved), the orientations of Ui​DU_{i}D and Si​DS_{i}D are also reversed (resp. preserved). Hence, the orientation of ΓD​(π)\Gamma_{D}(\pi) is reversed (resp. preserved). By the induction hypothesis, σk−1\sigma_{k-1} vanishes on orientation relations. Since σk\sigma_{k} is the sum of ΛD​(π)=σk−1​ΓD​(π)\Lambda_{D}(\pi)=\sigma_{k-1}\Gamma_{D}(\pi) for π∈𝔖k​(d)\pi\in\mathfrak{S}_{k}(d), σk\sigma_{k}(orientation relations)=0=0.

(Σ​3\Sigma 3) Let DI+DH+DXD_{I}+D_{H}+D_{X} be an IHX relation. Then ΓDI​(π)+ΓDH​(π)+ΓDX​(π)\Gamma_{D_{I}}(\pi)+\Gamma_{D_{H}}(\pi)+\Gamma_{D_{X}}(\pi) also satisfies an IHX relation, because the procedure does not change the part IHX occurs. Since σk−1\sigma_{k-1} satisfies (Σ​3\Sigma 3) by the induction hypothesis,

ΛDI​(π)+ΛDH​(π)+ΛDX​(π)\displaystyle\Lambda_{D_{I}}(\pi)+\Lambda_{D_{H}}(\pi)+\Lambda_{D_{X}}(\pi)
=\displaystyle= σk−1​(ΓDI​(π)+ΓDH​(π)+ΓDX​(π))\displaystyle\sigma_{k-1}(\Gamma_{D_{I}}(\pi)+\Gamma_{D_{H}}(\pi)+\Gamma_{D_{X}}(\pi))
=\displaystyle= 0.\displaystyle 0.

(Σ​4\Sigma 4) Let Si​D=D−Ui​DS_{i}D=D-U_{i}D be an STU relation. Then we have

σk​(D−Ui​D)\displaystyle\sigma_{k}(D-U_{i}D)
=\displaystyle= 1|𝔖k​(D)|​∑π∈𝔖k​(D)(ΛD​(π)−ΛUi​D​(π))\displaystyle\frac{1}{|\mathfrak{S}_{k}(D)|}\sum_{\pi\in\mathfrak{S}_{k}(D)}(\Lambda_{D}(\pi)-\Lambda_{U_{i}D}(\pi))
=\displaystyle= 1|𝔖k​(D)|​∑π∈𝔖k​(D)(ΛD​(π)−(ΛD​(π​Ui)−ΛD​(Ui)))\displaystyle\frac{1}{|\mathfrak{S}_{k}(D)|}\sum_{\pi\in\mathfrak{S}_{k}(D)}(\Lambda_{D}(\pi)-\left(\Lambda_{D}(\pi U_{i})-\Lambda_{D}(U_{i})\right))
=\displaystyle= ΛD​(Ui)=σk−1​ΓD​(Ui)=σk−1​(Si​D)=σk​(Si​D).\displaystyle\Lambda_{D}(U_{i})=\sigma_{k-1}\Gamma_{D}(U_{i})=\sigma_{k-1}(S_{i}D)=\sigma_{k}(S_{i}D).

∎

4.2 ι:𝒜B​C​Rc→𝒜B​C​R\iota:\mathcal{A}^{c}_{BCR}\rightarrow\mathcal{A}_{BCR} is an isomorphism

Proof of Theorem 2.4.

We construct the inverse κ∗\kappa_{\ast} (resp. κ¯∗\overline{\kappa}_{\ast}) of the map ι∗\iota_{\ast} (resp. ι¯∗\overline{\iota}_{\ast}). Namely, we construct the map κ:𝒟→𝒜B​C​Rc\kappa:\mathcal{D}\rightarrow\mathcal{A}^{c}_{BCR} (resp. κ:𝒟→𝒜¯B​C​Rc\kappa:\mathcal{D}\rightarrow\overline{\mathcal{A}}^{c}_{BCR}) which vanishes on STU, IHX and orientation relations (and chord relations). Note that by Lemma 4.2 below, an IHX relation can be written a sum of several STU relations. Let DD be a (colored) BCR graph. By using STU relations repeatedly, we can transform the graph DD to a linear combination H=∑kak​DkH=\sum_{k}a_{k}D_{k}, where each DkD_{k} is a BCR chord diagram. We call this operation resolution of DD. We set κ​(D)\kappa(D) as the equivalence class of HH in 𝒜B​C​Rc\mathcal{A}^{c}_{BCR}. By construction, the map κ\kappa, if it is well-defined, induces κ∗:𝒜B​C​R→𝒜B​C​Rc\kappa_{\ast}:\mathcal{A}_{BCR}\rightarrow\mathcal{A}^{c}_{BCR}. We can easily check κ∗​ι∗=i​d\kappa_{\ast}\iota_{\ast}=id and ι∗​κ∗=i​d\iota_{\ast}\kappa_{\ast}=id.

However, we must show that the resulting H=H​(D)H=H(D) does not depend, up to 4​T4T relations, on choices of procedures of STU relations. We show this by induction on the number of internal vertices #​W​(D)\#W(D).

The case #​W​(D)=0\#W(D)=0 is trivial because no STU relation is performed. Let #​W​(D)=1\#W(D)=1, and let ww be the unique internal vertex. Since ww has three adjacent dashed edges e1e_{1}, e2e_{2} and e3e_{3}, there are three choices of STU relations. Each choice produces a sum of two (or one) chord diagrams. We can easily see that the differences between these combinations are exactly 4​T4T relations.

Next, we assume that the case #​W​(D)≤i\#W(D)\leq i is proved and show the case #​W​(D)=i+1\#W(D)=i+1 (i≥1)(i\geq 1). Let an edge e1e_{1} of the graph DD connect an external vertex v1v_{1} and an internal vertex w1w_{1}. Let another edge e2e_{2} of DD connect an external vertex v2v_{2} and an internal vertex w2w_{2}. First, we assume w1≠w2w_{1}\neq w_{2}. Consider the two procedures to perform STU relations. One performs the first STU relation to e1e_{1} and gives a sum of graphs D1​t+D1​uD_{1t}+D_{1u}, while another performs the first STU relation to e2e_{2} and gives D2​t+D2​uD_{2t}+D_{2u}. Note that after the first choice of STU relation, the resulting combination of chord diagrams does not depend, up to 4​T4T relations, on the remaining choices of STU relations. Then we can assume that the second STU relation of D1​t+D1​uD_{1t}+D_{1u} is performed to the edge e2e_{2} (we write this procedure as e1→e2e_{1}\rightarrow e_{2}), while the second STU relation of D2​t+D2​uD_{2t}+D_{2u} is performed to the edge e1e_{1} (we write this as e2→e1e_{2}\rightarrow e_{1}). But the two procedures yield the same combination of graphs.

So the remaining case is the case w=w1=w2w=w_{1}=w_{2} and ww is connected to another internal vertex. If there is an edge e3e_{3} that connects external vertex v3v_{3} and an internal vertex w3≠ww_{3}\neq w, the procedure (e1→e3)(e_{1}\rightarrow e_{3}) is equal to (e3→e1)=(e3→e2)=(e2→e3)(e_{3}\rightarrow e_{1})=(e_{3}\rightarrow e_{2})=(e_{2}\rightarrow e_{3}). So let ww be connected to a block BB which consists of only internal vertices (and dashed edges). We see in Lemma 4.3 that in both cases (e1e_{1} first and e2e_{2} first), the result of STU relations to BB after the STU relation to eie_{i} vanishes by 4​T4T relation.

This ends the proof of well-definedness of κ\kappa. Finally, we show that the map κ\kappa sends chord relations to chord relations. Let ∑iDi(=0)\sum_{i}D_{i}(=0) be a chord relation such that D1D_{1} has an internal vertex w1w_{1} connected by an edge e1e_{1} to an external vertex v1v_{1}. Consider performing the STU relation to e1e_{1} and transform D1=D1​sD_{1}=D_{1s} to D1​t+D1​uD_{1t}+D_{1u}. Since graphs DiD_{i} are related by chord relations, there are corresponding edges ei=(vi,wi)e_{i}=(v_{i},w_{i}) and corresponding STU relations Di=Di​s=Di​t+Di​uD_{i}=D_{is}=D_{it}+D_{iu} for each DiD_{i}. We can observe that ∑iDi​t\sum_{i}D_{it} and ∑iDi​u\sum_{i}D_{iu} are chord relations.

∎

4.3 IHX in terms of STU

Lemma 4.2.

Recall that 𝒟\mathcal{D} is the space generated by BCR graphs with at least one external vertex. Then an IHX relation in 𝒟\mathcal{D} can be written as a sum of STU relations.

Proof..

Let a graph DD have an “II” part, namely two internal vertices w1w_{1} and w2w_{2} connected by a dashed edge ee. Consider the IHX relation performed on the edge ee. We have three graphs D=D1D=D_{1}, D2D_{2} and D3D_{3}. We want to show D1+D2+D3=0D_{1}+D_{2}+D_{3}=0 by only using STU relations.

Using STU relation simultaneously to D1D_{1}, D2D_{2} and D3D_{3}, we can assume that w1w_{1} is connected to an external vertex vv by some dashed edge ff. Then perform the STU relation to ff and ee in this order. This yields four or two graphs for each DiD_{i}, depending on the number of solid edges the external vertex vv has. We can easily see that these twelve (or six) graphs are canceled. See Figure 10.

Refer to caption
Figure 10: IHX is a sum of STU relations

∎

4.4 Sliding of chords

Lemma 4.3 (Sliding of chords).

Let DD be a BCR chord diagram. Let w1w_{1} be an external vertex and let w1w_{1} have one dashed edge (w0,w1)(w_{0},w_{1}) that is connected to an external vertex w0w_{0}. Let v1,v2,…,v2​nv_{1},v_{2},\dots,v_{2n} be other external vertices. Assume that these vertices satisfy the following. See Figure 11.

  • •

    w1w_{1} is connected by one solid edge ee to v1v_{1}

  • •

    v1,v2,…,v2​nv_{1},v_{2},\dots,v_{2n} are connected by nn chords among them.

  • •

    viv_{i} and vi+1v_{i+1} are connected by one solid edge.

  • •

    viv_{i} has no other solid edge except for at most one solid edge connected to v2​nv_{2n}. We write this exceptional edge as ff.

Refer to caption
Figure 11: Example of sliding argument

We construct another chord diagram D′D^{\prime} from DD by remove the solid edge e=(v1,w1)e=(v_{1},w_{1}) and inserting a new solid edge e′=(v2​n,w1)e^{\prime}=(v_{2n},w_{1}) to ff.

Then by using 4​T4T relation, DD and D′D^{\prime} are equivalent.

Proof..

4T relation can be written as D1+D2+D3+D4=0D_{1}+D_{2}+D_{3}+D_{4}=0, where Di​(i=1,2,3,4)D_{i}(i=1,2,3,4) are four graphs obtained by four ways to connect an end w1w_{1} of dashed edge to the sides of some fixed chord. Consider the sum of 4​n4n graphs obtained by all the ways to connect w1w_{1} to the sides of the nn chords. By 4​T4T relation, this sum is equal to 0. On the other hand, observe that in the sum there are 2​n−12n-1 pairs that consist of the two isomorphic graphs with the opposite sign. (See Figure 12.)The remaining graphs are DD and D′D^{\prime}.

Refer to caption
Figure 12: Four graphs D1D_{1}, D2D_{2}, D3D_{3}, D4D_{4} (above). A pair of isomorphic graphs with the opposite sign (below).

∎

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