Morse theory for group presentations &
applications to the persistent fundamental group
XIMENA FERNANDEZ
Durham University
UK CENTRE FOR TOPOLOGICAL DATA ANALYSIS
Applied Topology Seminar - CIMAT
'Every mathematician has a secret weapon.
Mine is Morse theory.'
Raoul Bott.
Combinatorial Homotopy Theory
Combinatorial Homotopy Theory
Let K,L be CW-complexes.
- There is an elementary collapse K↘eL (or elementary expansion Le↗K)
Combinatorial Homotopy Theory
Let K,L be CW-complexes.
- A collapse K↘L is an ordered pairing {(ei,xi):1≤i≤k} of the cells in K∖L such that if Ki=L∪⋃j<iej∪⋃j<ixj, then Ki↘eKi−1.
Simple Homotopy Theory
Let K,L be CW-complexes.
- There is an n-deformation K↗↘nL if there is a sequence of CW-complexes K=K0,K1,…Kr=L such that Ki↘eKi+1 or Kie↗Ki+1 for each 0≤i≤r−1, and dim(Ki)≤n for all 1≤i≤r.
- There is simple homotopy equivalence K↗↘L if K↗↘nL for some n∈N.
Simple Homotopy Theory
Let K,L be CW-complexes.
- K↗↘L⇒K≃L.
- The converse is not true, the obstruction is measured by the Whitehead group.
If the Whitehead group Wh(K) of the complex K is trivial, then any complex homotopy equivalent to K is also simple homotopy equivalent to K. - For K contractible, Wh(K)=0 and hence K≃∗⇔K↗↘∗.
- (Whitehead '50) If K,L are n-complexes, then K↗↘L⇒K↗↘n+1L if n≠2.
For n=2, this question is open. - (C.T.C. Wall '66) If K,L are 2-complexes, then K↗↘L⇒K↗↘4L.
Discrete Morse Theory
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
- f:K→R a discrete Morse function.
For every cell en in K, #{en≻en−1:f(en)≤f(en−1)}≤1 and #{en≺en+1:f(en)≥f(en+1)}≤1.
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
- f:K→R a discrete Morse function.
- C the set of critical cells of K.
An n-cell en∈K is a critical cell of index n if the values of f in every face and coface of en increase with dimension.
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
- f:K→R a discrete Morse function.
- C the set of critical cells of K.
Theorem [Forman, '98]. K is homotopy equivalent to a CW-complex KM with exactly one cell of dimension k for every critical cell of index k.
Discrete Morse Theory & Whitehead deformations
Discrete Morse Theory & Whitehead deformations
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪˜en where the attaching map ˜φ:∂Dn→L of ˜en is defined as ˜φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Discrete Morse Theory & Whitehead deformations
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪˜en where the attaching map ˜φ:∂Dn→L of ˜en is defined as ˜φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Discrete Morse Theory & Whitehead deformations
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪˜en where the attaching map ˜φ:∂Dn→L of ˜en is defined as ˜φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Discrete Morse Theory & Whitehead deformations
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪˜en where the attaching map ˜φ:∂Dn→L of ˜en is defined as ˜φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Discrete Morse Theory & Whitehead deformations
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪˜en where the attaching map ˜φ:∂Dn→L of ˜en is defined as ˜φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Discrete Morse Theory & Whitehead deformations
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪˜en where the attaching map ˜φ:∂Dn→L of ˜en is defined as ˜φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Discrete Morse Theory & Whitehead deformations
Proposition [F. 2017]: Let K∪d⋃i=1ei be a CW-complex where
dim(K)≤dim(ei)≤dim(ei+1)≤n for all i=1,2,…,d.
Let φj:∂Dj→K∪⋃i<jei be the attaching map of ej.
If K↘L, then there exist CW-complexes Z1≤Z2≤⋯≤Zd of dim ≤n+1 such that for every j=1,2,…,d,
K∪j⋃i=1ei↗Zj↘L∪j⋃i=1˜ei where the attaching map
˜φj:∂Dj→L∪⋃i<j˜ei of the cell ˜ej is defined inductively as:
∙ ˜φ1=r0φ1 with r0:K→L the canonical strong deformation retract and for j>1,
∙ ˜φj=˜rj−1ıj−1φj where ˜rj−1:Zj−1→L∪⋃i<j˜ei is the strong deformation retract and ıj−1:K∪⋃i<jei→Zj−1 is the inclusion.
Discrete Morse Theory & Whitehead deformations
Theorem [F. 2017]: Let K be a regular CW-complex of dim n and let f:K→R be discrete Morse function. Then, f induces a sequence of internal collapses given by a filtration of K ∅=K−1⊆L0⊆K0⊆L1⊆K1⋯⊆LN⊆KN=K such that Kj↘Lj for all 1≤j≤N and Lj=Kj−1∪dj⋃i=1eji with {eji:0≤j≤N,1≤i≤dj} the set of critical cells of f. Moreover, K↗↘n+1L0∪N⋃j=1dj⋃i=1˜eji=KM.
* Here, the attaching maps of the cells ˜eji can be explicitly reconstructed from the internal collapses.
Discrete Morse Theory & Whitehead deformations
Morse theory for
group presentations
CW-complexes of dim 2
Group presentations
CW-complexes of dim 2
KP
Group presentations
P=⟨x,y | xyx−1y−1⟩
CW-complexes of dim 2
K
Group presentations
PK=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16, x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Morse theory for group presentations
K
PK=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16, x−110x9x4,x−19x8,x−18x7x6,x−17⟩
K+f:K→R Morse function
PKM=⟨x4,x6 | x6x4x−16x−14⟩
Morse theory for group presentations
Given K a regular CW-complex of dim 2 and f:K→R a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation PKM.
O(M2) with M = |2-cells of K|
ximenafernandez/Finite-Topological-Spaces (SAGE)
Posets-Package (GAP) (joint with K. Piterman & I. Sadofschi Costa).
(c.f. Brendel, Ellis, Juda, Mrozek. Fundamental group algorithm for low dimensional tessellated CW-complexes, 2015. arXiv:1507.03396)
Example
Q0=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
ω0(x13,x−113x12x−14)=x12x−14
Example
Q0=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
ω0(x13,x−113x12x−14)=x12x−14
Q1=⟨x4,x6,x7,x8,x9,x10,x11,x12 | x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
ω1(x12,x−112x12)=x11
Q2=⟨x4,x6,x7,x8,x9,x10,x11 | x11x−14,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Example
Q0=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Q1=⟨x4,x6,x7,x8,x9,x10,x11,x12 | x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Q2=⟨x4,x6,x7,x8,x9,x10,x11 | x11x−14,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Q3=⟨x4,x6,x7,x8,x9,x10 | x10x−16x−14,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Q4=⟨x4,x6,x7,x8,x9 | x9x4x−16x−14,x−19x8,x−18x7x6,x−17⟩
Q5=⟨x4,x6,x7,x8 | x8x4x−16x−14,x−18x7x6,x−17⟩
Q6=⟨x4,x6,x7 | x7x6x4x−16x−14,x−17⟩
Q7=⟨x4,x6 | x6x4x−16x−14⟩
Morse theory for group presentations
Given K a regular CW-complex of dim 2 and f:K→R a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation PKM.
The Andrews-Curtis conjecture
The Andrews-Curtis conjecture
Conjecture [Andrews & Curtis, 1965]. Any finite balanced presentation P=⟨x1,…,xn | r1,…,rn⟩ of the trivial group can be transformed into the empty presentation ⟨ | ⟩ by a finite sequence of the following operations:
- replace some relator ri by r−1i
- replace some relator ri by rirj for some j≠i
- replace some relator ri by a conjugate wriw−1 for some w in the free group F(x1,x2,…,xn)
- add a generator xn+1 and a relator that coincides with xn+1, or the inverse of this operation
Tietze transformations
Theorem [Tietze, 1908]. Any finite presentation P=⟨x1,…,xn | r1,…,rm⟩ of a group G can be transformed into any other presentation of the same group by a finite sequence of the following operations:
- replace some relator ri by r−1i
- replace some relator ri by rirj for some j≠i
- replace some relator ri by a conjugate wriw−1 for some w in the free group F(x1,x2,…,xn)
- add a generator xn+1 and a relator that coincides with xn+1, or the inverse of this operation
- add a relator 1, or the inverse of this operation
The Andrews-Curtis conjecture
Potential counterexamples.
- P=⟨x,y | xyx=yxy, xn=yn+1⟩, n≥2 [Akbulut & Kirby, 1985]
- P=⟨x,y | x−1ynx=yn+1, x=y−1xyx−1⟩, n≥2 [Miller & Schupp, 1999]
- P=⟨x,y | x=[xm,yn],y=[yp,xq]⟩, n,m,p,q∈Z [Gordon, 1984]
Theorem [Bridson, 2015] There exist balanced presentations of the trivial group that satisfy the AC-conjecture for which the minimun length of any simplification sequence is superexponential in the total length of the relators.
The Andrews-Curtis conjecture
Remark [Referee]. AC-transformations of group presentations are in correspondence with 3-deformations of 2-complexes.
Morse theory for group presentations
Corollary [F. 2017]. Let P be a finite presentation of a finitely presented group. Let K′P be the barycentric subdivision of the standard complex KP and let f:K′P→R be a discrete Morse function.
Then P∼ACP(K′P)M.
The Andrews-Curtis conjecture
Theorem [F. 2017-2021]. The following balanced presentations of the trivial group satisfies the Andrews-Curtis conjecture:
∙ P=⟨x,y | xyx=yxy, x2=y3⟩* [Akbulut & Kirby, 1985]
∙ P=⟨x,y | x−1y3x=y4, x=y−1xyx−1⟩ [Miller & Schupp, 1999]
∙ P=⟨x,y | x=[x−1,y−1],y=[y−1,xq]⟩,∀q∈N [Gordon, 1984]
* First proved by Miasnikov in 2003 using genetic algorithms.
The Persistent fundamental group
Topological inference
Let Xn be a (possible noisy) finite sample of an unknown topological space X.
Topological inference
Let Xn be a (possible noisy) finite sample of an unknown topological space X.
Goal: Describe (algorithmic) tools to infer topological properties of X from Xn.
Topological inference
Let Xn be a (possible noisy) finite sample of an unknown topological space X.
Goal: Describe (algorithmic) tools to infer topological properties of X from Xn.
Topological inference
Let Xn be a (possible noisy) finite sample of an unknown topological space X.
Goal: Describe (algorithmic) tools to infer topological properties of X from Xn.
Persistence
- Point cloud Xn
Persistence
- Point cloud Xn
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
Persistence
- Point cloud Xn
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
- Apply a topological invariant F F(K1)(i1)∗→F(K2)(i2)∗→⋯(iN−1)∗→F(KN)
Persistence
- Point cloud Xn
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
- Apply a topological invariant F F(K1)(i1)∗→F(K2)(i2)∗→⋯(iN−1)∗→F(KN)
- For every i<j, there exist a homomorphism fi,j:F(Ki)→F(Kj) and the persistent invariant i→j is defined as Fi→j:=Im(fi,j)
The persistent fundamental group
- Point cloud Xn
- Filtration of (connected) complexes K1i1→K2i2→⋯iN−1→KN
- Apply π1 π1(K1)(i1)∗→π1(K2)(i2)∗→⋯(iN−1)∗→π1(KN)
- For every i<j, there exist a homomorphism fi,j:π1(Ki)→π1(Kj) and the persistent fundamental group i→j is defined as πi→j1:=Im(fi,j)
- Algorithm?
The persistent fundamental group
Morse reduction. Given f:KN→R a Morse function (K1)M?→(K2)M?→⋯?→(KN)M
π1((K1)M)?→π1((K2)M)?→⋯?→π1((KN)M)
The persistent fundamental group
- Point cloud Xn
- Filtration (connected with constant set of vertices) K1i1→K2i2→⋯iN−1→KN
- f:KN→R a Morse function (with a single critical 0-cell)
- Persistent fundamental group
We developed an algorithm with complexity O(M2) with M = |2-cells of KN| to compute presentations of π1((Kj)M) and homomorphisms gj:π1((Kj)M)→π1((Kj+1)M) for all j s.t. π1(K1)(i1)∗→π1(K2)(i2)∗→⋯(iN−1)∗→π1(KN) ≅ ↺ ~~~~~~~~~~\pi_1((K_1)_\mathcal{M})\xrightarrow{g_1} \pi_1((K_2)_\mathcal{M}) \xrightarrow{g_2} \dots \xrightarrow{g_{N-1}}\pi_1((K_N)_\mathcal{M})
The Persistent fundamental group
~O(M^2) with M = |2-cells of K_N|
The Persistent fundamental group
~O(M^2) with M = |2-cells of K_N|
Example
S^2\vee S^1\vee S^1 vs S^1\times S^1
Example
S^2\vee S^1\vee S^1 vs S^1\times S^1
Example
S^2\vee S^1\vee S^1 vs S^1\times S^1
Vietoris–Rips filtration
Persistent homology
Example
S^2\vee S^1\vee S^1 vs S^1\times S^1
Persistent fundamental group
For \alpha \in [1, \sqrt{2}) and \alpha' \in [\sqrt{2}, 2), \varphi_{\alpha, \alpha'}(x_i) = \begin{cases} x_{16} & \text{if }i = 13,14,15,16\\ x_{17} & \text{if }i = 4,8,12,17\\ 1 & \text{otherwise}.\end{cases}
Example
S^2\vee S^1\vee S^1 vs S^1\times S^1
Persistent fundamental group
Persistent homology
Future work
- Persistence of group invariants.
- Graphical representation (interaction with Persistent Homology).
- Canonical Morse functions for Vietoris-Rips filtrations of point clouds.
- Stability to small perturbations.
- Implementation in GAP & SAGE.
- Applications in real data.
References
- PhD thesis. X. Fernandez, Combinatorial methods and algorithms in low-dimensional topology and the Andrews-Curtis conjecture. University of Buenos Aires, 2017.
- Article. X. Fernandez. Morse theory for group presentations. Transactions of the AMS (to appear) arXiv:1912.00115, 2019.
- Github repository. ximenafernandez/Finite-Topological-Spaces (SAGE)
- Tutorial. Morse Theory 2.0 at AATRN Youtube Channel.
Thanks!
github.com/ximenafernandez ~~~~ ximenafernandez.github.io/ ~~~~ ximena.l.fernandez@durham.ac.uk ~~~~ @pi_ene
Example
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle~~~~~~P_{(K_2)_\mathcal M} =\langle x_4, x_6, x_7~ | ~ x_7x_6 x_4x_6^{-1}x_4^{-1}\rangle
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle~~~~~~P_{(K_2)_\mathcal M} =\langle x_4, x_6, x_7~ | ~ x_7x_6 x_4x_6^{-1}x_4^{-1}\rangle ~~~~~~ P_{(K_3)_\mathcal M} = \langle x_4, x_6~ | ~ x_6 x_4x_6^{-1}x_4^{-1}\rangle
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle