# Growth Series of the Braid MonoidMB5 in Band Generators

Authors: Muhammad Haleem Khan; Zaffar Iqbal
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

Growth series is an important invariant associated with group or monoid which classifies all the words of group or monoid. Therefore, the growth series of braid monoids and Hecke algebras in Artin’s generators is presented in many scholarly published articles. The growth series of braid monoidsMB3 and MB4 in band generators is known. In this work, we compute the complete presentation of braid monoid MB5 in band generators by solving all the ambiguities of MB5. The words on the left-hand of each relation are reducible words, and the words on the right-hand side are canonical words. We partially find the growth series Q∗5 of reducible words. Then, we construct a linear system for canonical words of MB5 in band presentation and compute the corresponding growth series. We also find the growth rate of growth series of MB5 in band generators.


---

## Body

## 1. Introduction

The growth series also known as Hilbert series is an important invariant in the study of modern geometry. In physics, growth series have recently become a power full tool in high energy theory, appearing, for example in the study of Bogomol’nyi-Prasad-Sommerfield operators of supersymmetric gauge theories [1, 2]; supersymmetric quantum chromodynamics [3, 4], and instanton moduli space [5, 6]. In [7], Hilbert series was used to construct an operator basis in 1/m expansion of a theory with a nonrelativistic heavy fermion in an electromagnetic (NRQED) or color gauge field (NRQCD/HQET).The braid groupBn+1 admits the presentation given by Artin [8].
(1)Bn+1=x1,⋯,xnxixj=xjxi,ifi−j≥2xi+1xixi+1=xixi+1xi,if1≤i≤n−1.The braid groupBn+1 admits other presentations such as Sergiescu graph-presentation and Birman-Ko-Lee presentation or band presentation. The last presentation is given by
(2)Bn+1=ats,n≥t>s≥1atsarq=arqats,t−rs−rs−qt−q>0atsasr=atrats=asratr,n≥t>s>r≥1.Growth series of braid monoidMB3 and MB4 is computed in [9]. In [10], growth series of the finite dimensional Hecke algebra is presented. Growth series for graded S-module was computed by Haider in [11]. The growth series of binomial edge ideals was computed by Kumar and Sarkar in [12]. In [13], growth series of the graded algebra of real regular functions on the symplectic quotient associated to an SU2-module is computed and given an explicit expression for the first nonzero coefficient of the Laurent expansion of the growth series at t=1. Growth series and the coefficient of Laurent expansion of growth series of special linear group of 2×2 matrices are computed in [14]. In [15], Saito proved that the growth functions associated with Artin monoids of finite type are rational functions whose nominator is 1 and the denominator is polynomial Nnt having distinct roots. Growth series of symplectic quotients by 2-torus is computed in [16]. In [17], the Laurent coefficients of the growth series of a Gorenstein algebra is presented. In the growth series 1/qnt of MBn (for n=3,4,5,6) for Artin generators, the degrees of the polynomials qnt are 3, 4, 10, 15, respectively (for detail see [18]). Universal upper bound for the growth of Artin monoid is computed in [19]. Growth series of braid monoid MB3 and MB4 in band generators is computed in [9]. The degrees of polynomial qnt in case of band generators are 2 and 3 (for n=3 and 4). In this paper, we compute the growth series of braid monoid MB5 using band presentation, and we see that the degree of the polynomial qnt is 4. We note that growth rate of MB5 in band generators is much slower than that of Artin generators.


## 2. Materials and Methods

InMB5, we fix a total order a21<a31<a32<⋯<ann−1 on the generators. In the monoid, the relation α=β will be written as α<β in the length-lexicographic order. Let α1=uw and α2=wv; then the word of the form uwv is said to be ambiguity (for detail see [20]). If α1v=uα2 as a relation as well as in the length-lexicographic order, then, we say that the ambiguity is solvable. A presentation is complete if and only if all the ambiguities are solvable (for detail see [21, 22]). Corresponding to the relations α=β, the changes γαδ⟶γβδ give a rewriting system. A presentation will be called a complete presentation if and only if all the ambiguities are solvable.In a complete presentation (or in the general presentation) ofMBn+1if a word W containsα, then W is called a reducible word and we denote it by B∗n in general. If word W does not containα, then W is called canonical word or canonical form. Let U and V be nonempty words; then, the word UaijV will be denoted as aij×ijaijV.Definition 1 (see [23]).
LetG be a finitely generated group and S be a finite set of generators of G. Then, the word length lsg of an element g∈G is the smallest integer n for which there exist s1,⋯,sn∈SUS−1 such that g=s1,⋯,sn.Definition 2 (see [23]).
LetG be a finitely generated group and S be a finite set of generators of G. Then, the growth function of the pair G,S associates to an integer k≥0 the number ak of the element g∈G such that lsg=k, and the corresponding growth series is given by PGt=∑k=0∞aktk.In 2008, Bokut [22] gave the Gro¨bner-shirshov basis (GSB) of Bn+1 in band generators. The notion of this basis is in [14, 20, 24–27] under different names: complete presentation, presentations with solvable ambiguities, Gro¨bner-shirshov basis, rewriting system, and so on. In [28], we proved the subset of GSB of Bn+1 given by Bokut [26] is a GSB of MBn+1. Using the notation (used in [28]) t,s for generator ats and Vt,s or Wt,s for the words in akl such that t≥k>l≥s, we have.Theorem 3 (see [28]).
A GSB of braid monoidMBn+1 consist of following relations:
(3)k,li,j=i,jk,lk>l>i>j,k,lVi−1,1i,j=i,jVj−1,1k,lk>i>j>l,t3,t2t2,t1=t2,t1t3,t1,t3,t1Vt2−1,1t3,t2=t2,t1t3,t1Vt2−1,1,t,sVt2−1,1t2,t1Wt3−1,t1t3,t1=t3,t2t,sVt2−1,1t2,t1Wt3−1,t1′,t3,sVt2−1,1t2,t1Wt3−1,t1t3,t1=t2,st3,sVt2−1,1t2,t1Wt3−1,t1′,Fort3>t2>t1,t>t3,t2>sandWt3−1,t1t3,t1=t3,t1Wt3−1,t1′,where Wt3−1,t1′=Wt3−1,t1p,q⟶p,q,ifq≠t1;p,t1⟶t3,p.Proposition 4 (see [9]).
Solution of linear system for canonical words of braid monoidMB4 is given by
(4)P214=t1−t1−5t+5t2,P314=t1−5t+5t2,P324=t1−5t+5t2,P414=t1−2t1−5t+5t2,P424=t−t21−5t+5t2,P434=t1−2t1−5t+5t2.
And the growth series ofMB4 in band generators is given as
(5)PM4t=11−t1−5t+5t2.


## 3. Results and Discussion

In this paper, we compute the growth series ofMB5 (in band presentation). From Equation (2), we have the following band-presentation of MB5:
(6)a54,a53,a52,a51,a43,a42,a41,a32,a31,a21R13,R23,Ri4,Rj5,i=3,⋯,10,j=11,⋯,30,where
(7)R13:a31a32=a21a31,R23:a32a21=a21a31,R34:a41a32=a32a41,R44:a41a42=a21a41,R54:a41a43=a31a41,R64:a42a21=a21a41,R74:a42a43=a32a42,R84:a43a21=a21a43,R94:a43a31=a31a41,R104:a43a32=a32a42,R115:a51a32=a32a51,R125:a51a42=a42a51,R135:a51a43=a43a51,R145:a51a52=a21a51,R155:a51a53=a31a51,R165:a51a54=a41a51,R175:a52a43=a43a52,R185:a52a21=a21a51,R195:a52a53=a32a52,R205:a52a54=a42a52,R215:a53a21=a21a53,R225:a53a31=a31a51,R235:a53a32=a32a52,R245:a53a54=a43a53,R255:a54a21=a21a54,R265:a54a31=a31a54,R275:a54a32=a32a54,R285:a54a41=a41a51,R295:a54a42=a42a52,R305:a54a43=a43a53are given basic relations.For the braid monoidMB5, we have given another form of Theorem 3 that is directly used to compute the growth series of MB5. This form is obtained by solving all the ambiguities in the band presentation of MB5.Proposition 5.
A complete presentation ofMB5 for band presentation is given by
(8)a54,a53,a52,a51,a43,a42,a41,a32,a31,a21R13,R23,R843,Ri4,Rj4,Rk5,i=3,⋯,10,j=76,⋯,83,k=11,⋯,75,
where the new relationsR315,… ,R843 are given as follows
(9)R315:a51a21na31=a32a51a21a32n−1,R325:a51a21na41=a42a51a21a42n−1,R335:a51a21na32W32a41=a42a51a21a42n−1W′32,R345:a51a21na32ra42=a43a51a21na32a43r−1,R355:a51a21na43=a43a51a21n,R365:a51a21na51=a21a51a21a52n−1,R375:a51a21na32W32a51=a21a51a21a52n−1W′32,R385:a51a21na42V42a51=a21a51a21a52n−1V′42,R395:a51a21na32W32a43V43a51=a21a51a21a52n−1W′32,21V′43,R405:a51a21na32ra31W31a42V42,21a51=a21a51a21a52n−1a32rW′31V′42,R415:a51a21na32ra52=a31a51a21na32a53r−1,R425:a51a21na32ra43V43,42,32a52=a31a51a21na32a53r−1V′43,42,32,R435:a51a21na32ra31W31a42V42,43,32a52=a41a51a21na32rW′31V′42,43,32,R445:a51a21na42V42,43,32a52=a41a51a21nV′42,43,32,R455:a51a21na53=a31a51a21n,R465:a51a21na32W32a43ra53=a41a51a21nW′32a43r,R475:a51a21na54=a41a51a21n,R485:a51a21na32W32a54=a41a51a21nW′32,R495:a51a31W31a41=a43a51a31W′31,R505:a51a31W31a51=a31a51a31W′31,R515:a51a31W31a42V42a51=a31a51a31W′31V′42,R525:a51a31W31a43V43a51=a31a51a31W′31V′43,R535:a51a31W31a42V42,43,32a52=a41a51a31W′31V′42,43,32,R545:a51a31W31a43na53=a41a51a31W′31a43a54n−1,R555:a51a31W31a54=a41a51a31W′31,R565:a51a41V41a51=a41a51a41V′41,R575:a52a31W31a41=a43a52a31W′31,R585:a52a31W31a51=a32a52a31W′31,R595:a52a31W31a42V42,a51=a32a52a31W′31V′42,R605:a52a31W31a43V42a51=a32a52a31W′31V′43,R615:a52a31W31a42V42,43,32a52=a42a52a31W′31V′42,43,32,R625:a52a31W31a43na53=a42a52a31W′31a43a54n−1,R635:a52a31W31a54=a42a52a31W′31,R645:a52a32na42=a43a52a32a43n−1,R655:a52a32W32a41V41a51=a42a52a32W′32V′41,R665:a52a32na52=a32a52a32a53n−1,R675:a52a32na31W31a42V42,43,32a52=a42a52a32nW′31V′42,43,32,R685:a52a32na43V43,42,32a52=a32a52a32a53n−1V′43,42,32,R695:a52a32W32a43na53=a42a52a32W′32a43a54n−1,R705:a52a32W32a54=a42a52a32W′32,R715:a52a41V41a51=a42a52a41V′41,R725:a52a42V42,43,32a52=a42a52a42V′42,43,32,R735:a53a41V41a51=a43a53a41V′41,R745:a53a42V42,43,32a53=a43a53a42V′42,43,32,R755:a53a43na53=a43a53a43a54n−1,R764:a41a21na31=a32a41a21a32n−1,R774:a41a21na41=a21a41a21a42n−1,R784:a41a21na32ra42=a31a41a21na32a43r−1,R794:a41a21na43=a31a41a21n,R804:a41a21a32W32a41=a21a41a21a32W′32,R814:a41a21a31W31a41=a31a41a31W′31,R824:a42a31W31a41=a32a42a31W′31,R834:a42a32na42=a32a42a32a43n−1,R843:a31a21na31=a21a31a21a32n−1,where n and r are positive integers, W3k a canonical word in MB3 starting with a3k,k=1,2 and
(i)
W′3k=W3k: a32⟶a32,a21⟶a41,a31⟶a43,V4l is a canonical word in MB4 starting with a4l,l=1,2,3(ii)
V′3l=V3l:a41⟶a54,a42⟶a42,a43⟶a53,a32⟶a32,a21⟶a41,a31⟶a43 and V4m,32=∑A3m−13,m=2,3 (as mentioned in Theorem 3)Proof.
We denote the ambiguity formed by left sides of the relationRi and Rj in MB5 by Ri-Rj=swt (say). If in the ambiguity z=swt,Lz=swt and Rz=swt are different lexicographically, then we get a new relation in the complete presentation of MB5, and if Lz=swt and Rz=swt are reduced to an identical word, then we say the ambiguity is solvable and no new relation is formed. The above relations are formed by solving the ambiguities involving basic relations and new relations.
For an ambiguityR115−R23=a51a32a21=w1(say), we have
(10)Rw1=a51a32a21¯=a51a21a31,Lw1=a51a32¯a21=a32a51a21.
Hence, we have a new relationRw15:a51a21a31=a32a51a21. Again, by solving new ambiguity Rw15−R13=a51a21a31a32=w2(say), we have
(11)Rw2=a51a21a31a32¯=a51a212a31,Lw2=a51a21a31¯a32=a32a51a21a32.
Hence, we have another relationRw25:a51a212a31=a32a51a21a32. Now by solving ambiguity Rw25−R13=a51a212a31a32=w3(say), we have
(12)Rw3=a51a212a31a32¯=a51a213a31,Lw2=a51a212a31¯a32=a32a51a21a322.
Hence, we have a relationRw25:a51a213a31=a32a51a21a322. By continuing the same process, we have the general relation
(13)R315:a51a21na31=a32a51a21a32n−1,n≥1.
For an ambiguityR125−R64=a51a42a21=w4 (say), we have
(14)Rw4=a51a42a21¯=a51a21a41,Lw4=a51a42¯a21=a42a51a21.
Hence, we have a new relationRw45:a51a21a41=a42a51a21. Again, by solving new ambiguity Rw45−R44=a51a21a41a42=w5 (say), we have
(15)Rw5=a51a21a41a42¯=a51a212a41,Lw5=a51a21a41¯a42=a42a51a21a42,which give another relation Rw55:a51a212a41=a42a51a21a42. Now by solving ambiguity Rw55−R43=a51a212a41a42=w6(say), we have
(16)Rw6=a51a212a41a42¯=a51a213a41,Lw6=a51a212a41¯a42=a42a51a21a422.
Hence,Rw65:a51a213a41=a42a51a21a422. By continuing the same process, we have the general relation
(17)R325:a51a21na41=a42a51a21a42n−1,n≥1.
For an ambiguityR125−R74=a51a42a43=w7 (say), we have
(18)Rw7=a51a42a43¯=a51a32¯a42=a32a51a42¯=a32a42a51,Lw7=a51a42¯a43=a42a51a43¯=a42a43¯a51=a32a42a51,where Lw7=Rw7 are reduced to identical word, so ambiguity is solvable and no new relation is formed. Using a similar procedure, we obtained all above new relations in complete presentation of MB5. Hence the proof is omitted.As defined above,A∗m denotes the set of canonical words and B∗m the set of reducible words in MBn+1 in general. In particular, B5i:kl:mn:pq;rs5, where 1≤i≤3 denote the set of reducible words starting with a5iakl and ending on ars. In this notation arsis a generator in MBd: 3≤d≤5, pqdenotes the canonical word (possibly empty) in MBd−1 starting with apq, mn denotes the canonical word (possibly empty) in MBd−2 starting with amn, and kl denotes the canonical word (possibly empty) in MBd−3 starting with akl. We will denote the empty word by ϕ. We denote the set all reducible words starting with a5iakl and ending on ars by B5i.kl;rs5 . Hence, B5i:kl:mn:pq;rs5 is a subset of B5i.kl;rs5.We are using other notions as follows:(i)
We denote the seta21,a212,a213,⋯ by A212(ii)
Aijn denotes the set of canonical words starting with aij in MBn(iii)
∑rAijn denotes the set of all the word in Aijn such that the index of each generator is increased by r. Hence, ∑rAijn=Aijn, i.e., the cardinality remains unchanged. In particular for the set A212=a21,a212,a213,⋯, we have ∑A212=a32,a322,a323,⋯(iv)
Anj.kln denotes the set of canonical words starting with anjakl in MBn(v)
The growth series ofB∗m, A∗m and MB5 is denoted by Q∗m,P∗m and PM5t, respectivelyNote that growth series ofA212 is P212=t/1−t.Proposition 6.
The following equalities hold for the reducible words inMB5:
(19)Q51.21;315=t31−t.Q51.21;415=t31−2t,Q51.21;425=t41−t2,Q51.21;435=t31−t,Q51.21;515=t3−2t41−5t+5t2,Q51.21;525=t42−6t+5t21−t21−2t2,Q51.21;535=t31−3t+3t21−t21−2t,Q51.21;545=t31−2t,Q51.31;415=t31−2t,Q51.31;515=t3−t41−5t+5t2,Q51.31;525=t41−t2,Q51.31;535=t41−t1−2t,Q51.31;545=t31−2t,Q51.41;515=t31−2t1−5t+5t2,Q52.31;415=t31−2t,Q52.31;515=t3−t41−5t+5t2,Q52.31;525=t41−t2,Q52.31;535=t41−t1−2t,Q52.31;545=t31−2t,Q52.32;425=t31−t,Q52.32;515=t41−5t+5t2,Q52.32;525=t31−3t+3t21−t1−2t2,Q52.32;535=t41−t1−2t,Q52.32;545=t31−2t,Q52.41;515=t31−2t1−5t+5t2,Q52.42;525=t31−2t,Q53.41;515=t31−2t1−5t+5t2,Q53.42;525=t31−2t.Q53.43;535=t31−t.Proof.
Using simple decomposition of words and⨆ denotes the disjoint union of sets then, we have
(1)
SinceB51.21;315=B51:21;315=a51a21na31 = a51×A212×a31, hence , Q51.21;315=t3/1−t(2)
B51.21;415=B51:21:ϕ;415⨆B51:21:32;415 =a51a21na41⨆a51a21na32V32a41=a51×A212×a41⨆a51×A212×A323×a41 implies Q51.21;415=t3/1−2t(3)
B51.21;425=B51:21:32;425=a51a21na32ra42=a51×A212×∑A212×a42 implies Q51.21;425=t4/1−t2
Using similar procedure, we obtained all aboveQ∗5 of MB5. Hence, the proof is omitted.Next, we construct linear system for canonical form inMB5.Proposition 7.
The following equalities hold for the canonical words inMB5:
(20)P215=t1−t1−5t+5t21+∑i=14P5i5,P315=t1−5t+5t21+∑i=14P5i5,P325=t1−5t+5t21+∑i=14P5i5,P415=t1−2t1−5t+5t21+∑i=14P5i5,P425=t−t21−5t+5t21+∑i=14P5i5,P435=t1−2t1−5t+5t21+∑i=14P5i5,P515=t+tP515+∑i=24P51.i15,P525=t+t∑i=12P5i5+∑i=12P52.3i5+∑i=12P52.4i5,P535=t+t∑i=13P5i5+∑i=13P53.4i5,P545=t+t∑i=14P5i5,P51.215=tP215−t21−tP315−t21−2tP415−t31−t2P425−t21−tP435−t2−2t31−5t+5t2P515−t32−6t+5t21−t21−2t2P525−t21−3t+3t21−t21−2tP535−t21−2tP545,P51.315=tP315−t21−2tP415−t2−t31−5t+5t2P515−t31−2t2P525−t31−t1−2tP535−t21−2tP545,P51.415=tP415−t21−2t1−5t+5t2P515,P52.315=tP315−t21−2tP415−t2−t31−5t+5t2P515−t31−2t2P525−t31−t1−2tP535−t21−2tP545,P52.325=tP325−t21−tP425−t31−5t+5t2P515−t21−3t+3t21−t1−2t2P525−t31−t1−2tP535−t21−2tP545,P52.415=tP415−t21−2t1−5t+5t2P515,P52.425=tP425−t21−2tP525,P53.415=tP415−t21−2t1−5t+5t2P515,P53.425=tP425−t21−2tP525,P53.435=tP435−t21−tP535.Proof.
We compute the growth series inductively. By using decomposition of words, we have(1)
A215=A214⨆A214×A515⨆A525⨆A535⨆A545 gives P215=t/1−t1−5t+5t21+∑i=14P5i5(2)
A315=A314⨆A314×A515⨆A525⨆A535⨆A545 implies P315=1/1−5t+5t21+∑i=14P5i5(3)
A325=A324⨆A324×A515⨆A525⨆A535⨆A545 implies P325=1/1−5t+5t21+∑i=14P5i5(4)
A415=A414⨆A414×A515⨆A525⨆A535⨆A545 implies P415=t1−2t/1−5t+5t21+∑i=14P5i5(5)
A425=A424⨆A424×A515⨆A525⨆A535⨆A545 implies P425=t−t2/1−5t+5t21+∑i=14P5i5(6)
A435=A434⨆A434×A515⨆A525⨆A535⨆A545 implies P435=t1−2t/1−5t+5t21+∑i=14P5i5
The setA5i5 consists of all the words starting with the generator a5i. Therefore, the set a5i×A5i5 is subset of A5i5 consisting of all the words starting with a5i2. We apply this concept in the proof of items (7), (8), (9), and (10).
(7)
The setA515 is disjoint union of its subsets a51,a51×A515,A51.215,A51.315,A51.415, i.e A515=a51⨆a51×A515⨆A51.215⨆A51.315⨆A51.415. Therefore, we have P515=t+tP515+∑i=24P51.i15
Similarly, we have(8)
A525=a52⨆a52×A525⨆a52×A515⨆A52.315⨆A52.325⨆A52.415⨆A52.425 implies P525=t+t∑i=12P5i5+∑i=12P52.3i5+∑i=12P52.4i5(9)
A535=a53⨆a53×A535⨆a53×A525⨆a53×A515⨆A53.415⨆A53.425⨆A53.435 impliesP535=t+t∑i=13P5i5+∑i=13P53.4i5(10)
A545=a54⨆a54×A545⨆a54×A535⨆a54×A525⨆a54×A515 implies P545=t+t∑i=14P5i5(11)
A51.215=a51×A215B51.21;315×31A315⨆B51.21;415×41A415⨆B51.21;425×42A425⨆B51.21;435×43A435⨆B51.21;515×51A515⨆B51.21;525×52A525⨆B51.21;535×53A535⨆B51.21;545×54A545 implies(21)P51.215=tP215−t21−tP315−t21−2tP415−t31−t2P425−t21−tP435−t2−2t31−5t+5t2P515−t32−6t+5t21−t21−2t2P525−t21−3t+3t21−t21−2tP535−t21−2tP545(12)
A51.315=a51×A315/B51.31;415×41A415⨆B51.31;515×51A515⨆B51.31;525×52A525⨆B51.31;535×53A535⨆B51.31;545×54A545 implies(22)P51.315=tP315−t21−2tP415−t2−t31−5t+5t2P515−t31−2t2P525−t31−t1−2tP535−t21−2tP545(13)
A51.415=a51×A415\B51.41;515×51A515 implies P51.415=tP415−t21−2t/1−5t+5t2P515(14)
A52.315=a52×A315B52.31;415×41A415⨆B52.31;515×51A515⨆B52.31;525×52A525⨆B52.31;535×53A535⨆B52.31;545×54A545 implies(23)P52.315=tP315−t21−2tP415−t2−t31−5t+5t2P515−t31−2t2P525−t31−t1−2tP535−t21−2tP545(15)
A52.325=a52×A325B52.32;425×42A425⨆B52.32;515×51A515⨆B52.32;525×52A525⨆B52.32;535×53A535⨆B52.32;545×54A545 implies(24)P52.325=tP325−t21−tP425−t31−5t+5t2P515−t21−3t+3t21−t1−2t2P525−t31−t1−2tP535−t21−2tP545(16)
A52.415=a52×A415\B52.41;515×51A515 implies P52.415=tP415−t21−2t/1−5t+5t2P515(17)
A52.425=a52×A425\B52.42;525×52A525 implies P52.425=tP425−t2/1−2tP525(18)
A53.415=a53×A415\B53.41;515×51A515 gives P53.415=tP415−t21−2t/1−5t+5t2P515(19)
A53.425=a53×A425\B53.42;525×52A525 implies P52.425=tP425−t2/1−2tP525(20)
A53.435=a53×A435\B53.43;535×53A535 implies P53.435=tP435−t2/1−tP535Theorem 8.
The Hilbert series of the braid monoidMB5 in band generators is given by
(25)PM5t=11−t1−2t1−7t+7t2.Proof.
Solving the system of linear equations constructed in Proposition7, we get
(26)P215=t1−t1−2t1−7t+7t2,P315=t1−2t1−7t+7t2,P325=t1−2t1−7t+7t2,P415=t1−7t+7t2,P425=t1−7t+7t2,P435=t1−7t+7t2,P515=t−5t2+5t31−2t1−7t+7t2,P525=t−t21−7t+7t2,P535=t−t21−7t+7t2,P545=t−5t2+5t31−2t1−7t+7t2.
Therefore, the Hilbert series of the braid monoidMB5 is given by
(27)PM5t=1+P215+∑i=12P3i5+∑i=13P4i5+∑i=14P5i5=11−t1−2t1−7t+7t2.Remark 9.
By partial fractions of growth series ofMB5 we have 1/1−t1−2t1−7t+7t2=−1/1−t−8/31−2t−7/31−7−21/2t+7/31−7+21/2t. The only term that contributes in approximation of the series is 7/31−7+21/2t and 7/31−7+21/2t=7/31+7+21/2t+7+21/22t2+⋯. Therefore, the growth function is ak5=7/37+21/2k, and hence, the growth rate of MB5 is 7+21/2(approximately equal to 5.791).


## 4. Conclusion

From band presentation of braid monoidMB5 we solve all the ambiguities and get new and more interesting relations R315 to R845 which are given in Proposition 5. When no more ambiguity is remaining to be solved, then these new relations with the basics relations are complete presentation of braid monoid MB5 in band generators. The words on the left-hand side of these new relations are reducible words, and the words on the right-hand side are canonical words. In Proposition 6, we partially find the growth series Q∗5 of reducible words. We also construct a linear system for canonical words. The most important outcome of our work is the growth series of braid monoid MB5 in band generators which is given in Theorem 8. These results are very interesting and useful for mathematicians. Using these results, one can find the growth series of higher order braid monoids and can generalize the results for MBn.



---

*Source: 9108480-2022-06-07.xml*