Throughout this section, we assume that i) graph \(\Gamma \) is finite, connected and has at least one essential vertex, ii) \(\Gamma \) is sufficiently subdivided with respect to a fixed \(n>1\). This means that the edge-length of every path between two essential vertices is at least \(n-1\) and the edge-length of every loop is at least \(n+1\).
Morse presentations
We first review Farley-Sabalka’s algorithm [10] which provides a group presentation of the graph braid group for any graph in terms of cells of an appropriate Morse complex. Certain features of Morse presentations of graph braid groups will be key ingredients of some of our proofs in later subsections. Farley-Sabalka’s algorithm realises a specific instance of Forman’s discrete Morse theory [25] on the discrete configuration space, \(D_n(\Gamma )\). A discrete configuration space is a regular cube complex that is a deformation retract of \(C_n(\Gamma )\). For the deformation-retracion \(C_n(\Gamma )\rightarrow D_n(\Gamma )\) to work, \(\Gamma \) has to be sufficiently subdivided with respect to the particle number n [26, 27]. Cells of \(D_n(\Gamma )\) have the following form
$$\begin{aligned} c=\{c_1,c_2,\ldots ,c_n\}\subset \Gamma \mid {\overline{c}}_i\cap {\overline{c}}_j=\emptyset \quad \forall i\ne j \end{aligned}$$
where \(\{c_i\}_{i=1}^n\) are mutually disjoint closed cells of \(\Gamma \), i.e. they are either edges or vertices. Cell c can be viewed as a cube
Hence the dimension of c is given by \(|E(\Gamma )\cap c|\). In this way, the discrete configuration space can also be viewed as a proper subset of \(C_n(\Gamma )\).
The next step is the construction of a rooted spanning tree that determines the ordering of vertices of \(\Gamma \). We choose a rooted spanning tree \((T,*)\subset \Gamma \) such that \(*\) is univalent in T. Edges that belong to the complement of T in \(\Gamma \) are called deleted edges. We assume that T is chosen so that the endpoints of deleted edges are always bivalent vertices.
We fix a planar embedding \(T\subset {\mathbb {R}}^2\) and label vertices in T as follows.
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The root \(*\) is labelled by 0.
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To determine labels of the remaining vertices, we imagine a thin ribbon neighbourhood of T. Starting from \(*\) and travelling clockwise along the boundary of the ribbon, we label the visited vertices as \(1,2,\ldots , |V|-1\).
We denote the deleted edge that is adjacent to \(*\) by \(e_0\) (if such an edge exists). Between any two vertices \(\{v_1,v_2\}\subset V(\Gamma )\) there exists a unique edge path from \(v_1,v_2\) in T. Such a path will be denoted by \([v_1,v_2]\). Therefore, by choosing a rooted spanning tree, we have a linear order on \(V(\Gamma )\) that we denote by <. For any edge \(e\in E(\Gamma )\) we define its initial and terminal vertices denoted by \(\iota (e)\) and \(\tau (e)\) respectively so that \(\iota (e)>\tau (e)\). Similarly, for any \(v\in V(\Gamma )\) we define e(v) as the edge for which \(\iota (e(v))=v\). For an embedding between labelled graphs \(f:\Gamma _1\rightarrow \Gamma _2\), we say that f is order-preserving if it preserves the order of vertices.
Example 2
For a subdivided \(\Theta _k\), we may choose a rooted spanning tree \((T_k,*)\) as follows:
Then the ribbon of \(T_k\) and labels are given below.
In the description of the graph braid group, we are particularly interested in loops and paths in \(C_n(\Gamma )\). Note first that any path in \(C_n(\Gamma )\) with its endpoints contained in \(D_n(\Gamma )\subset C_n(\Gamma )\) is homotopy equivalent to a path that is entirely contained in \(D_n(\Gamma )\). Furthermore, any path in \(D_n(\Gamma )\) is homotopy equivalent to a path that is fully contained in the 1-skeleton of \(D_n(\Gamma )\). In particular, any element \({\mathbf {B}}_n(\Gamma )\) can be represented as a word of 1-cells on \(D_n(\Gamma )\). More precisely, cell \(\{e,v_1,\ldots ,v_{n-1}\}\) is viewed as a directed path from \(\{\iota (e),v_1,\ldots ,v_{n-1}\}\) to \(\{\tau (e),v_1,\ldots ,v_{n-1}\}\).
The general idea of the discrete Morse theory is to contract configuration space \(D_n(\Gamma )\) to a much smaller Morse complex \(M_n(\Gamma )\) whose 1-skeleton is a wedge of circles based at the same point given by configuration \(\{0,1,\ldots ,n-1\}\). Then, one uses the fact that \({\mathbf {B}}_n(\Gamma )\cong \pi _1\left( M_n(\Gamma )\right) \), i.e. \({\mathbf {B}}_n(\Gamma )\) is generated by the circles in the 1-skeleton of \(M_n(\Gamma )\).
The construction of discrete Morse theory relies on the notion of Morse matching. A Morse matching is a collection of partially defined maps \(\{W_d\}_{d=0}^{n}\) where \(W_i\) maps some of d-cells in \(D_n(\Gamma )\) to some of \((d+1)\)-cells. In Farley-Sabalka’s algorithm the Morse matching can be understood in terms of imposing certain rules for particle movement on \(\Gamma \). A d-dimensional cell \(c\in D_n(\Gamma )\) is viewed as a move where d particles simultaneously slide along edges \(e\in E(\Gamma )\cap c\) in the direction from \(\iota (e)\) to \(\tau (e)\). We say that an edge \(e\in c\) is \(order-respecting \) if \(e\subset T\) and there are no vertices \(v\in c\) for which \(v<\tau (e)\) and \(\tau (e(v))=\tau (e)\). Intuitively, on junctions in \(\Gamma \) only particles that have no other particles to their right (in terms of vertex ordering on T) are allowed to move first. In a similar spirit one defines blocked vertices as vertices whose movement is blocked in c by some other particles. Vertex \(v\in c\) is blocked if there exists \(c_i\in c\) such that \(\tau (e(v))\cap c_i\ne \emptyset \). Clearly, any edge \(e\in c\) is either order-respecting or non order-respecting. Similarly, \(v\in c\) can be either blocked or unblocked.
Definition 1
A cell \(c\in D_n(\Gamma )\) is critical if and only all vertices \(v\in V(\Gamma )\cap c\) are blocked and all edges \(e\in E(\Gamma )\cap c\) are non order-respecting.
The definition of map \(W_d\) is recursive. Namely, the domain of \(W_d\) consists of cells that do not belong to the image of \(W_{d-1}\) and are not critical. For such cells, \(W_d\) finds the lowest unblocked vertex in c, \(v_0\), and replaces it with edge e(v), i.e.
$$\begin{aligned}W_d(c)=\left( c\setminus \{v_0\}\right) \cup \{e(v_0)\}.\end{aligned}$$
Cells that belong to the domain of some \(W_d\) are called redundant and cells that belong to the image of some \(W_{d}\) are called collapsible. The Morse complex \(M_n(\Gamma )\) is a complex whose d-skeleton consists of critical d-cells in \(D_n(\Gamma )\). Importantly, any d-cell c in can be mapped to \(M_n(\Gamma )\) as a word that consists only of critical cells. This is done by means of the principal reduction F. We will specify this construction only for 1-cells as this is the only relevant case for this paper. To this end, consider the boundary word of 2-cell \(W_1(c)\) for a redundant cell c and bring it to the form cw where w is a word of appropriate 1-cells. Then, we define the principal reduction of a redundant c as \(F(c)=w^{-1}\). The action of F is extended to critical sells as \(F(c)=c\) and to collapsible cells as \(F(c)=1\). We extend F to any word of 1-cells in a natural way. By applying map F to any word sufficiently many times, we always end up with a word that consists only of critical cells. Such a word is invariant under map F. Let us denote such a stable iteration of F by \(F^\infty \). We are now ready to state the central theorem of this subsection.
Theorem 2
( Farley-Sabalka). Group \({\mathbf {B}}_n(\Gamma )\) is generated by critical 1-cells in \(D_n(\Gamma )\). Relators are given by \(F^\infty (w)\) where w goes through boundary words of all critical 2-cells in \(D_n(\Gamma )\).
Recall that any critical 1-cell has the form \(c=\{e,v_1,\ldots ,v_{n-1}\}\) where e is a non order-respecting edge and all \(\{v_i\}_{i=1}^{n-1}\) are blocked. We will distinguish the following two types of critical 1-cells.
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1.
Critical cells associated with an essential vertex, i.e. \(e\in T\).
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2.
Critical cells associated with a deleted edge, i.e. \(e\in \Gamma \setminus T\).
A critical cell at an essential vertex has the following properties i) \(\tau (e)\) is an essential vertex and there exists i such that \(\tau (e(v_i))=\tau (e)\) and \(\tau (e)<v_i<\iota (e)\), ii) for every i, we have \(\tau (e(v_i))=v_j\) for some \(i\ne j\) or \(\tau (e(v_i))=\iota (e)\). Hence, the form of such a critical 1-cell can be uniquely determined by specifying i) the relevant essential vertex, ii) the distribution of particles on leaves incident at the essential vertex and iii) the leaf at v containing the non-order respecting edge in c. If the essential vertex is v and v is of valence k, such a critical cell will be denoted by \(v_i({\mathbf {b}})\), where \(i\in \{1,\ldots ,k-1\}\) is the leaf containing the non order-respecting edge and \({\mathbf {b}}=(b_1,b_2,\ldots ,b_{k-1})\) is a sequence of nonnegative integers that specifies the distribution of particles on leaves at vertex v.
Furthermore, the assumption that end-vertices of all deleted edges are bivalent implies that all critical cells that are associated with a deleted edge e are of the form \(\{e,0,1,\ldots ,n-2\}\) if \(\tau (e)\ne *\) and \(\{e,1,2,\ldots ,n-1\}\) otherwise.
Note also that under the above assumptions about the choice of the spanning tree, there is just one critical 0-cell in \(D_n(\Gamma )\), namely \(\{0,1,\ldots ,n-1\}\). Hence, the 1-skeleton of \(M_n(\Gamma )\) consists of 1-cells that are topological circles based at the unique 0-cell.
In the following subsections we establish a correspondence between geometric generators of \({\mathbf {B}}_n(\Gamma )\) and the above two types of critical 1-cells. Namely, critical cells at essential vertices will correspond to generators of the star type and critical cells associated with deleted edges will correspond to generators of the loop type.
Generators of the star type
One of the building blocks in geometric presentations of graph braid groups is an exhaustive description of \({\mathbf {B}}_n(\Gamma )\) when \(\Gamma =S_k\), a star graph with k leaves. This is because for any graph \(\Gamma \) generators of \({\mathbf {B}}_n(S_k)\) can be regarded as generators of \({\mathbf {B}}_n(\Gamma )\). To see this, for an essential vertex \(v\in V(\Gamma )\) of valency k consider the following order-preserving embedding of \(S_k\) into spanning tree \(T\subset \Gamma \) centred at v
where the central vertex of \(S_k\) is mapped to v, the 0-th edge of \(S_k\) is mapped to a path from \(*\) to v and the i-th edge of \(S_k\) (\(0<i<k\)) is mapped to the corresponding (subdivided) edge adjacent to v. For such an embedding, critical 1-cells of \(D_n(\Gamma )\) at v are in a one-to-one correspondence with critical 1-cells of \(D_n(S_k)\). Hence, \(\iota _v\) indices a well-defined map from \({\mathbf {B}}_n(S_k)\) to \({\mathbf {B}}_n(\Gamma )\).
In the remaining part of this subsection we will introduce geometric generators of the star type and by writing down all relations between them we will reproduce the following well-known result.
Theorem 3
[28]. The n-braid group \({\mathbf {B}}_n(S_k)\) of the star graph \(S_k\) is a free group of rank
The strategy for this subsection is to find for each critical 1-cell \(c\in D_n(S_k)\) its corresponding loop \(\gamma \subset C_n(\Gamma )\) such that the associated word of 1-cells, \(w_\gamma \), satisfies \(F^\infty (w_\gamma )=c\). To this end, we introduce a shorthand notation for the configuration where \(a_i\) points occupies ith leaf of \(S_k\). We denote the leaves by \(e_0,\ldots ,e_{k-1}\) and the respective particle configuration by \(e_0^{a_0}e_1^{a_1}\cdots e_k^{a_k}\).
We also denote by \(\beta ^i\) the path from configuration \({\mathbf {x}}e_0\) to configuration \({\mathbf {x}}e_i\) for some abstract configuration \({\mathbf {x}}\) where the top particle from leaf \(e_0\) is moved to leaf \(e_i\). Consequently, by \(\beta ^{-i}\) we denote the inverse of \(\beta ^i\).
Definition 2
(Y-exchange). For a sequence \({\mathbf {a}}=(a_1,\ldots , a_{i-1})\) with \(1\le a_j<k\), we define \(\beta ^{\mathbf {a}}\) as the concatenation
$$\begin{aligned} \beta ^{\mathbf {a}}= \beta ^{a_1}\cdots \beta ^{a_{i-1}}. \end{aligned}$$
For \(1\le a,b <k\), we define a loop, called a Y-exchange on leaves a, b as
$$\begin{aligned} \sigma _i^{{\mathbf {a}},a,b}:=\beta ^{{\mathbf {a}}}Y^{ab}\beta ^{-{\mathbf {a}}}:e_0^n\rightarrow e_0^n\in {\mathbf {B}}_n(S_k), \end{aligned}$$
where
$$\begin{aligned} -{\mathbf {a}}= (-a_{i-1},-a_{i-2},\ldots , -a_1), \end{aligned}$$
and \(Y^{ab}\) is the commutator of \(\beta ^a\) and \(\beta ^b\)
$$\begin{aligned} Y^{ab} := \beta ^{a,b,-a,-b}=\beta ^a\beta ^b\beta ^{-a}\beta ^{-b}. \end{aligned}$$
In particular,
$$\begin{aligned} \sigma _1^{a,b}=Y^{ab}. \end{aligned}$$
Remark 1
We will also use the notation \(\sigma _i^{\mathbf {a}}\) when \({\mathbf {a}}=(a_1,\ldots , a_{i+1})\) to denote \(\sigma _i^{{\mathbf {a}}',a_i,a_{i+1}}\) for \({\mathbf {a}}'=(a_1,\ldots , a_{i-1})\).
Remark 2
The Y-exchange move has been considered in many literatures. For example, Kurlin and Safi-Samghabadi in [29] described the motion of two objects having non-zero (but finite) size on a metric graph, which seems more realistic but is essentially the same as our situation.
Example 3
Paths \(\beta ^{1,3,1}\) and \(Y^{4,2}\) for respective starting configurations \(e_0^7\) and \(e_0^4e_1^2e_3^1\) are depicted in Fig. 1.
Roughly speaking, \(\sigma _i^{{\mathbf {a}},a,b}\) interchanges ith and \((i+1)\)st particles by using 0th, ath and bth edges after moving the first \((i-1)\) particles to leaves determined by sequence \({\mathbf {a}}\). Since \(\left( Y^{ab}\right) ^{-1}=Y^{ba}\) and \(Y^{aa}=1\), we have
$$\begin{aligned} \left( \sigma _i^{{\mathbf {a}},a,b}\right) ^{-1}&=\sigma _i^{{\mathbf {a}},b,a},&\sigma _i^{{\mathbf {a}},a,b}&=1. \end{aligned}$$
(3)
Remark 3
By counting the number of possible sequences \({\mathbf {a}}=(a_1,\ldots ,a_{i+1})\) for \(i=1,\ldots ,n-1\), we get that the total number of Y-exchanges in \({\mathbf {B}}_n(S_k)\) is
$$\begin{aligned} \sum _{i=1}^{n-1}(k-1)^{i-1}\left( {\begin{array}{c}k-1\\ 2\end{array}}\right) . \end{aligned}$$
Factors \((k-1)^{i-1}\) come from choosing \(i-1\) from \(k-1\) leaves to accommodate the first \(i-1\) particles. Factors \(\left( {\begin{array}{c}k-1\\ 2\end{array}}\right) \) stem from the choice of two leaves where the exchange of particles i and \(i+1\) takes place.
Next, we will proceed with the description of relations between the star-generators. To this end, we need to set up some more notation. Consider a path that connects configuration \(e_0^n\) with configuration \(e_0^{n-\sum b_j}e_1^{b_1}\cdots e_{k-1}^{b_{k-1}}\). Such a path will be unambiguously determined by the following sequence of nonnegative integers \({\mathbf {b}}=(b_1,\ldots , b_{k-1})\) that encodes the final particle configuration with \(b_i\) particles on leaf \(e_i\). Namely, we associate with \({\mathbf {b}}\) the following sequence
$$\begin{aligned} {{\overline{{\mathbf {b}}}}} = (\underbrace{k-1,k-1,\ldots , k-1}_{b_{k-1}},\underbrace{k-2,k-2,\ldots , k-2}_{b_{k-2}},\ldots ,\underbrace{1,1,\ldots , 1}_{b_1}). \end{aligned}$$
Path \(\beta ^{{{\overline{{\mathbf {b}}}}}}\) connects configuration \(e_0^n\) with configuration \(e_0^{n-\sum b_j}e_1^{b_1}\cdots e_{k-1}^{b_{k-1}}\) as desired. Moreover, when written as a word of 1-cells of \(D_n(\Gamma )\), \(\beta ^{{{\overline{{\mathbf {b}}}}}}\) consists only of collapsible cells. To see this, note first that all 1-cells associated with one-particle moves in \(\beta ^{{{\overline{{\mathbf {b}}}}}}\), \(c=\{e,v_1,\ldots ,v_{n-1}\}\), are such that i) e is order-respecting and ii) vertex \(\iota (e)\) is the lowest unblocked vertex in the 0-cell \(c_0=\{\iota (e),v_1,\ldots ,v_{n-1}\}\). This in turn means that \(c=W_0\left( c_0\right) \), i.e. c is collapsible. Using similar arguments, one can check that a critical cell \(v_i({\mathbf {b}})\) for \({\mathbf {b}}=(b_1,\ldots ,b_{k-1})\) corresponds to the following concatenation of paths
$$\begin{aligned} v_i({\mathbf {b}}) = F^\infty \left( \beta ^{\overline{{\mathbf {b}}+e_i}}\cdot \beta ^{-i}\cdot \beta ^{-{{\overline{{\mathbf {b}}}}}}\right) , \end{aligned}$$
(4)
where
$$\begin{aligned} {\mathbf {b}}\pm e_i:=(b_1,\ldots , b_{i-1}, b_i\pm 1, b_{i+1},\ldots , b_{k-1}). \end{aligned}$$
To see this, note that the only critical cell in the word corresponding to 4 comes from the middle \(\beta ^{-i}\)-move and this is exactly critical cell \(v_i({\mathbf {b}})\).
Proposition 1
The set \(\{\sigma _i^{\mathbf {a}}\mid 1\le i\le n-1, {\mathbf {a}}=(a_1,\ldots , a_{i+1}), 1\le a_j\le k-1\}\) generates \({\mathbf {B}}_n(S_k)\).
Proof
Let us start with the simplest case of \(n=2\). Any critical 1-cell has the form \(\{e_i,w\}\) where \(e_i\) is the edge from ith leaf that is incident to the central vertex v and w is a vertex that is adjacent to v and \(v<w<\iota (e)\). According to our shorthand notation, such a 1-cell is denoted by \(v_i(w)\). Any two-particle exchange has the form \(\sigma _1^{a,b}=Y^{a,b}\). It is straightforward to see that
$$\begin{aligned}v_i(w)=F^\infty \left( Y^{\iota (e),w}\right) .\end{aligned}$$
In general, arbitrary critical cell \(v_i({\mathbf {b}})\in D_n(\Gamma )\) can be expressed as a word of a number of star generators. In the remaining part of the proof, we will construct an inductive way for finding such a word. In order to handle arbitrary n, for any \({\mathbf {b}}=(b_1,\ldots ,b_{k-1})\) consider the following splitting \({\mathbf {b}}_1:=(0,\ldots , 0,b_i, b_{i+1},\ldots , b_{k-1})\), \({\mathbf {b}}_2:=(b_1,\ldots , b_{i-1},0,\ldots , 0)\). From equation (4) we get that \(v_i({\mathbf {b}})\) is the following conjugation
$$\begin{aligned} v_i({\mathbf {b}})&\sim _{F^\infty } \beta ^{\overline{{\mathbf {b}}_1}} B \beta ^{-\overline{{\mathbf {b}}_1}},&B&=\beta ^i \beta ^{\overline{{\mathbf {b}}_2}} \beta ^{-i} \beta ^{-{{\overline{{\mathbf {b}}}}}_2}. \end{aligned}$$
Ley us next show that B is a word of Y-exchanges (modulo conjugation by appropriate paths) with the starting configuration being the end point of \(\beta ^{{{\overline{{\mathbf {b}}}}}_1}\), i.e. \(e_0^{n-b_i-\ldots -b_{k-1}}e_i^{b_i}\ldots e_{k-1}^{b_{k-1}}\). Indeed, for any sequence \({\mathbf {a}}=(a_1,\ldots ,a_\ell )\), we have the following:
$$\begin{aligned} \beta ^i\beta ^{{\mathbf {a}}}\beta ^{-i}\beta ^{-{\mathbf {a}}}=\beta ^i\beta ^{a_1}\beta ^{{\mathbf {a}}'}\beta ^{-i}\beta ^{-{\mathbf {a}}'}\beta ^{-a}=Y^{i,a_1}\cdot \beta ^{a_1}\left( \beta ^i\beta ^{{\mathbf {a}}'}\beta ^{-i}\beta ^{-{\mathbf {a}}'} \right) \beta ^{-a_1}, \end{aligned}$$
where \({\mathbf {a}}'=(a_2,\ldots , a_\ell )\). The above expression allows us to use the induction on the length of \({\mathbf {a}}\). Namely, by substituting \({\mathbf {a}}={{\overline{{\mathbf {b}}}}}_2\), we have
$$\begin{aligned} v_i({\mathbf {b}}) \sim _{F^\infty } \left( \beta ^{{{\overline{{\mathbf {b}}}}}_1}Y^{i,i-1}\beta ^{-{{\overline{{\mathbf {b}}}}}_1}\right) \cdot \left( \beta ^{{{\overline{{\mathbf {b}}}}}_1}\beta ^{i-1}\left( \beta ^i\beta ^{\overline{{\mathbf {b}}'}}\beta ^{-i}\beta ^{-\overline{{\mathbf {b}}'}}\right) \beta ^{-(i-1)}\beta ^{-{{\overline{{\mathbf {b}}}}}_1}\right) \\ =\sigma _{i_1}^{{{\overline{{\mathbf {b}}}}}_1,i,i-1}\cdot \left( \beta ^{{{\overline{{\mathbf {b}}}}}_1}\beta ^{i-1}\left( \beta ^i\beta ^{\overline{{\mathbf {b}}'}}\beta ^{-i}\beta ^{-\overline{{\mathbf {b}}'}}\right) \beta ^{-(i-1)}\beta ^{-{{\overline{{\mathbf {b}}}}}_1}\right) , \end{aligned}$$
where \(\overline{{\mathbf {b}}'}=(b_1,\ldots ,b_{i-1}-1)\) and \(i_1=1+b_{k-1}+b_{k-2}+\ldots +b_i\). By iterating the above inductive step for \(\beta ^i\beta ^{\overline{{\mathbf {b}}'}}\beta ^{-i}\beta ^{-\overline{{\mathbf {b}}'}}\), we get that \(v_i({\mathbf {b}})\) can be expressed as the \(F^\infty \)-image of a concatenation of the resulting star generators. \(\square \)
Proposition 2
There are relations among \(\sigma _i^{\mathbf {a}}\)’s as follows:
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1.
Pseudo-commutative relation: for \(|j-i|\ge 2\),
$$\begin{aligned} \sigma _j^{a_1\ldots a_{j+1}}\sigma _i^{a_1\ldots a_{i+1}} = \sigma _i^{a_1\ldots a_{i+1}}\sigma _j^{a_1\ldots a_{i-1}a_{i+1}a_ia_{i+2}\ldots a_{j+1}}. \end{aligned}$$
(5)
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2.
Pseudo-braid relation: for \(1\le i\le n-2\),
$$\begin{aligned}&\sigma _{i+1}^{a_1\ldots a_{i-1}a_i a_{i+1}a_{i+2}}\sigma _i^{a_1\ldots a_{i-1}a_i a_{i+2}}\sigma _{i+1}^{a_1\ldots a_{i-1}a_{i+2}a_i a_{i+1}}\nonumber \\&\quad =\sigma _i^{a_1\ldots a_{i-1}a_i a_{i+1}}\sigma _{i+1}^{a_1\ldots a_{i-1}a_{i+1}a_i a_{i+2}}\sigma _i^{a_1\ldots a_{i-1}a_{i+1}a_{i+2}}. \end{aligned}$$
(6)
Proof
The relations can be checked in a straightforward way by using the definition of star generators and expanding each generator as
$$\begin{aligned}\sigma _i^\mathbf{b}=\beta ^{b_1}\ldots \beta ^{b_{i-1}}Y^{b_i,b_{i+1}}\beta ^{-b_{i-1}}\ldots \beta ^{-b_1},\end{aligned}$$
where \(Y^{b_i,b_{i+1}}=\beta ^{b_i}\beta ^{b_{i+1}}\beta ^{-b_i}\beta ^{-b_{i+1}}\) and cancelling \(\beta ^x\beta ^{-x}\) whenever such an expression is encountered. \(\square \)
Note that relation (5) becomes trivial if \(a_i=a_{i+1}\) or \(a_j=a_{j+1}\). Similarly, relations (6) become trivial is at least one of the following equations is satisfied: \(a_i=a_{i+1}\), \(a_{i+1}=a_{i+2}\), \(a_i=a_{i+2}\). In particular, relations (6) are nontrivial if and only if \(k\ge 4\).
Let \(G_{n,k}\) be the abstract group whose generators are all \(\sigma _i^{{\mathbf {a}}}\) and relations are given by (5) and (6).
Lemma 1
Group \(G_{n,k}\) is a free group of rank \(N_1(n,k)\) that is generated by \(\sigma _i^{{\mathbf {a}}}\) such that \(1\le a_1\le a_2\le \ldots \le a_{i-1}\le a_{i+1}\) and \(a_{i}<a_{i+1}\).
Proof
We consider the group presentation of \(G_{n,k}\) having all \(\sigma _j^{\mathbf {a}}\) as generators and relations as in (5) and (6).
First, let us consider generators \(\sigma _{j}^{{\mathbf {a}}}\) with \(j=n-1\). Whenever \(a_{i+1}<a_i\) for some \(i\le n-3\), we can use a preudo-commutative relation (5) to swap \(a_{i+1}\) and \(a_i\). That is, we apply the following Tietze transformation
$$\begin{aligned}\sigma _{n-1}^{a_1\ldots a_{n}}=\sigma _i^{a_1\ldots a_{i+1}}\sigma _{n-1}^{a_1\ldots a_{i-1}a_{i+1}a_ia_{i+2}\ldots a_{n}}\left( \sigma _i^{a_1\ldots a_{i+1}}\right) ^{-1}\end{aligned}$$
to obtain a presentation of \(G_{n,k}\) where every \(\sigma _{n-1}^{{\mathbf {a}}}\) satisfies \(a_i\le a_{i+1}\). Applying analogous Tietze transformations for every \(i\in \{1,\ldots ,n-3\}\), we get a presentation of \(G_{n,k}\) where sequences \({\mathbf {a}}\) satisfy
$$\begin{aligned}a_1\le a_2\le \ldots \le a_{n-2}.\end{aligned}$$
Let us next show how pseudo-braiding relations (6) allow us to arrange triples \(a_{n-2}, a_{n-1}, a_n\). Whenever the triple \(a_{n-2}, a_{n-1}, a_n\) consists of pairwise distinct integers, relation (6) involves the following three generators
$$\begin{aligned} \sigma _{n-1}^{a_1\ldots a_{n-3} a_{n-2} a_{n-1} a_n},\ \sigma _{n-1}^{a_1\ldots a_{n-3} a_{n-1} a_n a_{n-2}}\quad \text { and }\quad \sigma _{n-1}^{a_1\ldots a_{n-3} a_{n-1} a_n a_{n-2}} \end{aligned}$$
and two more generators with \(j=n-2\). Note that the above three generators contain all permutations of \(a_{n-2}, a_{n-1}, a_n\). Therefore, no matter what magnitudes of these numbers are, one can always find an appropriate Tietze transformation that yields a presentation of \(G_{n,k}\) with
$$\begin{aligned} a_{n-2}&\le a_n,&a_{n-1}&<a_n. \end{aligned}$$
Summing up, using relations (5) and (6) we are able to obtain a presentation of \(G_{n,k}\) whose generators \(\sigma _{n-1}^{{\mathbf {a}}}\) are associated with sequences \({\mathbf {a}}\) such that \(1\le a_1\le a_2\le \ldots \le a_{n-2}\le a_n\) and \(a_{n-1}<a_n\). Let \(a_n=j\). Then the number of such sequences is \(\left( {\begin{array}{c}n+j-3\\ j-1\end{array}}\right) (j-1)\), where the factor \((j-1)\) comes from the choice of \(1\le a_{n-1}<j=a_n\). Therefore the total number of generators \(\sigma _{n-1}^{\mathbf {a}}\) is exactly
$$\begin{aligned} \sum _{j=1}^{k-1}\left( {\begin{array}{c}n+j-3\\ j-1\end{array}}\right) (j-1). \end{aligned}$$
It is straightforward to check that this number is the same as \(N_1(n,k)-N_1(n-1,k)\).
Finally, by applying analogous Tietze transformations for \(\sigma _{n-2}^{\mathbf {a}}, \sigma _{n-3}^{\mathbf {a}}\) and so on, we can reduce all relations and end up with exactly \(N_1(n,k)\) generators. \(\square \)
Theorem 4
The abstract group \(G_{n,k}\) is isomorphic to the braid group \({\mathbf {B}}_n(S_k)\) via the canonical map \(\phi :G_{n,k}\rightarrow {\mathbf {B}}_n(S_k)\) which sends \(\sigma _i^{\mathbf {a}}\) to an n-braid in \(S_k\) represented by \(\sigma _i^{\mathbf {a}}\).
Proof
By Propositions 1 and 2, map \(\phi \) is a surjective group homomorphism. Since both \(G_{n,k}\) and \({\mathbf {B}}_n(S_k)\) are free groups of the same rank \(N_1\), the surjective homomorphism \(\phi \) becomes an isomorphism since any finitely generated free groups are Hopfian. \(\square \)
The construction of star generators in terms of Y-exchanges can also be generalised so that it applies to any graph \(\Gamma \). This is done by considering an appropriate embedding of \(S_k\) into \(\Gamma \) as explained in the definition below.
Definition 3
For each essential vertex \(v\in V(\Gamma )\) of valency \(k\ge 3\) and a sequence \({\mathbf {a}}=(a_1,\ldots , a_{i+1})\) with \(1\le a_j\le k-1\), we define \(\sigma _i^{v;{\mathbf {a}}}\) by the image of \(\sigma _i^{{\mathbf {a}}}\in {\mathbf {B}}_n(S_k)\) under the map \((\iota _v)_*\)
$$\begin{aligned} \sigma _i^{v;{\mathbf {a}}} = (\iota _v)_*(\sigma _i^{{\mathbf {a}}}), \end{aligned}$$
where \(\iota _v\) is the order-preserving embedding
$$\begin{aligned} \iota _v:(S_k,*)\rightarrow (T,*)\subset (\Gamma ,*) \end{aligned}$$
such that \(\iota _v\) maps the central vertex of \(S_k\) to the chosen vertex v. We also denote \(\sigma _1^{v;a,b}\) by \(Y^{v;ab}\).
Generators of loop type
Generators of the loop type are in a one-to-one correspondence with deleted edges \(e\in \Gamma \setminus T\). To each deleted edge e we assign a unique loop in \(\Gamma \) using the path in T that joins \(\iota (e)\) and \(\tau (e)\). The form of such a loop is \(e\cup [\iota (e),\tau (e)]\) and we give it an orientation that agrees with the canonical orientation of e – from \(\iota (e)\) to \(\tau (e)\). We also distinguish one deleted edge \(e_0\) being the deleted edge which is adjacent to the root of T. Edge \(e_0\) will define a move which will be the counterpart of the total braid \(\delta \) in \({\mathbf {B}}_n({\mathbb {R}}^2)\). Other edges will correspond to one-particle moves. The base point for all moves is such that all particles are resting on the edge in T which is incident to the root \(*\). In terms of the corresponding Morse complex \(M_n(\Gamma )\), all loops are based at the point corresponding to the unique critical 0-cell \(\{0,1,\ldots ,n-1\}\). The loop generators are constructed as follows (see Fig. 2 an illustration of these generators).
-
1.
If \(e=e_0\), its corresponding loop is the following embedding of circle graph \(C_k\).
Generator \(\delta \) is the image of the generator of \({\mathbf {B}}_n(S^1)\) which takes the top particle around \(C_k\) and moves it to the bottom of the line. Generator \(\delta \) will be called the circular move. Such a move is unique (up to homotopy) provided that edge \(e_0\) exists.
-
2.
If \(e\ne e_0\), then there is an embedding \(\iota _e:Q\rightarrow \Gamma \), where Q is the lollipop graph
In this case, generator \(\gamma \) is the image of a particular generator of \({\mathbf {B}}_n(Q)\) which moves the top particle around \(\gamma \) while the remaining \((n-1)\) particles stay fixed on their positions on the edge which is the stick of the lollipop. We call \(\gamma \) a one-particle move. Then there are exactly \((b_1(\Gamma )-1)\) such generators where \(b_1(\Gamma )\) is the first Betti number of \(\Gamma \).
Let us next define another one-particle move \(\gamma _0\) for \({\mathbf {B}}_n(\Gamma )\) associated with edge \(e_0\). Since \(\Gamma \) is connected and contains at least one essential vertex, the loop represented by \(e_0\) must contain an essential vertex v. This means that we have a subgraph homeomorphic to the lollipop graph Q whose loop is represented by \(e_0\) and the lollipop’s stick is some edge incident at v that is not contained in the loop. We denote by \(e_{0}^v\) and \(e_{1}^v\) the two edges that are incident at v and lie on the loop in Q. Edge \(e_{0}^v\) is defined as the one which is closer to \(*\) than \(e_{1}^v\) in T. We denote the other edge incident at v by \(e_{2}^v\).
We define a braid \(\gamma _0\) which is a one-particle move defined as follows. We first move the top \(n-1\) particles to the edge \(e_{2}^v\) and then we move the remaining nth particle along the loop represented by \(e_0\). After the nth particle finishes the loop and returns to its original position, we move back the other \(n-1\) particles from \(e_{2}^v\) to their initial positions.
Then \(\gamma _0\) can be expressed as a word involving \(\sigma ^{v;{\mathbf {a}}}\), \(\gamma _0\) and \(\delta \). More specifically, Y-exchanges \(\sigma _i^{v;{\mathbf {a}}}\) at v satisfy the following relation
$$\begin{aligned} \gamma _0=\sigma _{n-1}^{v;2,\ldots ,2,1}\sigma _{n-2}^{v;2,\ldots ,2,1}\cdots \sigma _1^{v;2,1}\delta . \end{aligned}$$
(7)
Moreover, for any \({\mathbf {a}}=(a_1,\ldots ,a_{i+1})\) such that \(a_i\in \{1,2\}\) for all i, we have another relation
$$\begin{aligned} \sigma _{i+1}^{v;{\mathbf {a}}'}&=\delta \cdot \sigma _i^{v;{\mathbf {a}}}\cdot \delta ^{-1},&{\mathbf {a}}'&=(1,a_1,\ldots ,a_{i+1}). \end{aligned}$$
(8)
Both of the above relations can be verified by visualising their corresponding moves. LHS and RHS of the above equations are essentially the same moves up to some backtracking.
Remark 4
Let us next briefly argue how the above relations (7) and (8) will be used to relate graph braid groups to \({\mathbf {B}}_n({\mathbb {R}}^2)\). Namely, consider quotient group \({\mathbf {B}}_n(Q)/\langle \gamma _0\rangle \) which is generated only by star generators \(\sigma _i^{v;{\mathbf {a}}}\). By (7), circular move is then expressed as
$$\begin{aligned}\delta =\sigma _1^{v;1,2}\sigma _2^{v;2,1,2}\ldots \sigma _{n-1}^{v;2,\ldots ,1,2}.\end{aligned}$$
Substituting the above expression in equation (8), we get
$$\begin{aligned} \sigma _1^{v;1,2}\sigma _2^{v;2,1,2}X\sigma _1^{v;1,2}X^{-1}&= \sigma _2^{v;1,1,2}\sigma _1^{v;1,2}\sigma _2^{v;2,1,2}, \end{aligned}$$
(9)
where
$$\begin{aligned} X&=\sigma _3^{v;2,2,1,2}\sigma _4^{v;2,\ldots ,2,1,2}\cdots \sigma _{n-1}^{v;2,\ldots ,2,1,2}. \end{aligned}$$
If we forget all decorations such as v and \({\mathbf {a}}\), then we can use pseudo-braiding relations for \(S_3\) to show that X commutes with \(\sigma _1^{v;1,2}\). Then, equation (9) simplifies to the ordinary braid relation.
Generation of graph braid groups and connectivity
The following theorem summarises the material contained in previous sections.
Theorem 5
The graph braid group \({\mathbf {B}}_n(\Gamma )\) for arbitrary \(\Gamma \) is generated by the following moves
-
1.
Y-exchanges
$$\begin{aligned} \{\sigma _i^{v;{\mathbf {a}}}\mid 1\le i<n, v\in V, {\mathbf {a}}=(a_1,\ldots , a_{i+1}), 1\le a_j<{{\,\mathrm{val}\,}}(v)\}, \end{aligned}$$
-
2.
\(b_1(\Gamma )\)-many one-particle moves
$$\begin{aligned} \{\gamma _0,\ldots , \gamma _{b-1}\mid b=b_1(\Gamma )\}, \end{aligned}$$
-
3.
one circular move \(\delta \),
where \(b_1(\Gamma )\) is the first Betti number of \(\Gamma \).
Proof
This is a direct consequence of Farley-Sabalka’s algorithm and Proposition 1. Namely, any critical 1-cell associated with an essential vertex \(v\in \Gamma \) is \(F^\infty \)-equivalent to a word of Y-exchanges \(\sigma _i^{v;{\mathbf {a}}}\). Furthermore, any critical 1-cell associated with a deleted edge \(e\ne e_0\) is \(F^\infty \)-equivalent to a word of 1-cells corresponding to the loop generator associated with e. This is because such a word contains only collapsible cells and one critical cell which is precisely \(\{e,0,1,\ldots ,n-2\}\). By similar arguments, critical cell \(\{e_0,1,2,\ldots ,n-1\}\) is \(F^\infty \)-equivalent to the word of 1-cells associated with circular move \(\delta \). Hence, for every critical 1-cell in \(D_n(\Gamma )\), we have found its corresponding geometric generator. By theorem 2, such geometric moves generate \({\mathbf {B}}_n(\Gamma )\). \(\square \)
Our next result states that for 2-connected graphs the number of Y-exchanges needed to generate \({\mathbf {B}}_n(\Gamma )\) can be greatly reduced.
Proposition 3
Let \(\Gamma \) be a 2-connected graph. Then the braid group \({\mathbf {B}}_n(\Gamma )\) is generated by
$$\begin{aligned} \{Y^{v;ab}\mid v\in V, 1\le a<b<{{\,\mathrm{val}\,}}(v)\}\sqcup \{\gamma _0,\ldots ,\gamma _{b-1}\mid b=b_1(\Gamma )\}\sqcup \{\delta \}. \end{aligned}$$
Proof
It suffices to show that each \(\sigma _i^{v;{\mathbf {a}}}\) can be expressed as a word of \(Y^{w;ab}\), \(\gamma _j\), and \(\delta \). We use the induction on both i and the edge-length of the path \([*,v]\) in T. For \(i=1\), there is nothing to prove for any v.
Suppose that v is the very first essential vertex when counting from \(*\), i.e. there are no essential vertices in the interior of \([*,v]\). Let \(v'\) be the essential vertex which is the end of \(a_1\)th edge incident at v. Since \(\Gamma \) is 2-connected, \(\Gamma _v=\Gamma \setminus {{\,\mathrm{Star}\,}}(v)\) is connected. Therefore there exists a path \(\gamma \subset \Gamma _v\) from \(v'\) to \(*\). Such a path \(\gamma \) defines a loop \({{\overline{\gamma }}}\) based at \(*\) which is the union
.
We use loop \({{\overline{\gamma }}}\) to define a braid of the circular type which sends the top particle around \({{\overline{\gamma }}}\) to the last position, just as the circular move \(\delta \). We denote such a move by \({{\overline{\gamma }}}\) as well. Move \({{\overline{\gamma }}}\) is a word of \(\gamma _j^{\pm 1}\) and \(\delta \) so that
$$\begin{aligned} {{\overline{\gamma }}}=\gamma _{j_1}^{s_1}\ldots \gamma _{j_\ell }^{s_\ell }\delta , \end{aligned}$$
where \((j_1,\ldots ,j_\ell )\) is a sequence of indices of deleted edges that \(\gamma \) passes and \(s_k\in \{-1,1\}\), \(1\le k\le \ell \), are exponents coming from orientations of the deleted edges relative to the orientation of \(\gamma \). We also have a relation which is analogous to relation (8).
$$\begin{aligned} \sigma _i^{v;{\mathbf {a}}}&= {{\overline{\gamma }}}\sigma _{i-1}^{v;{\mathbf {a}}'}\left( {{\overline{\gamma }}}\right) ^{-1},&{\mathbf {a}}'&=(a_2,\ldots , a_{i+1}) \end{aligned}$$
By repeating the above inductive step for \(\sigma _{i-1}^{v;{\mathbf {a}}'}\) and another loop associated with edge \(e_{a_2}^v\) and so on, we end up with an expression that involves only one-particle moves, the circular move \(\delta \) and \(\sigma _1^{v;a_{i},a_{i+1}}\).
Now suppose that v is a vertex farther than the nearest vertices from \(*\) and \(i\ge 2\). As before, the connectivity of \(\Gamma _v\) implies the existence of a path \(\gamma \subset \Gamma _v\) from the \(v'\) to \(*\), where \(v'\ne v\) is the vertex which is the end point of the \(a_1\)th edge adjacent to v.
Suppose that \(\gamma \) does not intersect the path \([*,v]\) in T. Then the union
defines a loop \({{\overline{\gamma }}}\) based at \(*\) and by the exactly same argument as above, we have
$$\begin{aligned} \sigma _i^{v;{\mathbf {a}}} = {{\overline{\gamma }}}\sigma _{i-1}^{v;{\mathbf {a}}'}\left( {{\overline{\gamma }}}\right) ^{-1}, \end{aligned}$$
(10)
where
$$\begin{aligned} {\mathbf {a}}'&=(a_2,\ldots , a_{i+1}),&{{\overline{\gamma }}}&=\gamma _{j_1}^{s_1}\ldots \gamma _{j_\ell }^{s_\ell }\delta , \end{aligned}$$
and by the induction hypothesis, we are done.
Finally, suppose that \(\gamma \) intersects the path \([*,v']\) at \(w\ne *\). By taking the subpath of \(\gamma \), we may assume that \(\gamma \) is a composition of \(\gamma '\) and \([w,*]\) such that \(\gamma '\) is a path joining v and w and does not intersect \([*,v]\).
As before, it gives us loop \({{\overline{\gamma }}}=[*,v']\cup \gamma '\cup [w,*]\) based at \(*\). In this case, \({{\overline{\gamma }}}\) can be regarded as a one-particle move and expressed as a word
$$\begin{aligned} {{\overline{\gamma }}}=\gamma _{j_1}^{s_1}\ldots \gamma _{j_\ell }^{s_\ell }, \end{aligned}$$
where \((j_1,\ldots ,j_\ell )\) is the sequence of indices of deleted edges defined as the same as before and \(s_k\in \{-1,1\}\), \(1\le k\le \ell \) are the exponents.
Let us denote two edges adjacent to w contained in \([*,v]\) and \(\gamma '\) by \(e_a^w\) and \(e_b^w\), respectively. The conjugation \({{\overline{\gamma }}}^{-1}\sigma _i^{v;{\mathbf {a}}}{{\overline{\gamma }}}\) gives us a braid, which is a concatenation as follows:
-
1.
Move the first particle along \([*,w]\) to the edge \(e_b^w\).
-
2.
Move the next \((i-2)\) particles onto edges adjacent to v by using the sequence \((a_2,\ldots , a_{i-1})\).
-
3.
Interchange the positions of the next two particles by using the \(a_i\)th and \(a_{i+1}\)th edges adjacent to v and take them back to the edge at \(*\).
-
4.
Move the \((i-2)\) particles on edges of v back to the original position.
-
5.
Move the first particle back to the original position.
Then indeed, this is a conjugate of \(\sigma _{i-1}^{v;{\mathbf {a}}'}\) with \({\mathbf {a}}'=(a_2,\ldots , a_{i+1})\) by the braid \(\sigma \) which interchanges positions of the first particle with the next i particles at w by using edges \(e_a^w\) and \(e_b^w\), which is a word of Y-exchanges
$$\begin{aligned} \sigma =\sigma _1^{w;b,a}\sigma _2^{w;a,b,a}\ldots \sigma _{i}^{w;a,\ldots ,a,b,a}. \end{aligned}$$
Therefore we have
$$\begin{aligned} \sigma _i^{v;{\mathbf {a}}} = {{\overline{\gamma }}}\cdot \sigma \cdot \sigma _{i-1}^{v;{\mathbf {a}}'}\cdot \sigma ^{-1}\cdot {{\overline{\gamma }}}^{-1} \end{aligned}$$
(11)
By induction hypothesis, not only \(\sigma _{i-1}^{v;{\mathbf {a}}'}\) but also all \(\sigma _j^{w;-}\) can be expressed as words of \(\sigma _i^{w;-}\)’s since w is closer to \(*\) than v, which completes the proof. \(\square \)