(II) Consider the second homotopy move P0 = (xAByBAz) → (xyz) = P and its inverse move where (|A|,|B|) = (1,−1). It is necessary to consider two distinct cases (II-1), (II-2) as follows.
(II–1) Consider case where the state of P0 with (mark(A), mark(B)) = (1, 1) is represented as (u|
ABw
∅² |v).
S+−(², η) denotes the state (u| Aw
∅²| AB
∅η|v) of P0 with (mark(A), mark(B))
= (1, −1), S−+(², η) denotes the state (u| ABw
∅² | ABt
∅η|v) of P0 with (mark(A), mark(B)) = (−1, 1),S++(²) denotes the state (u|
ABw
∅² |v) ofP0with (mark(A), mark(B)) = (1, 1), andS−−(²) denotes the state (u|
ABw
∅² |v) ofP0with (mark(A), mark(B)) = (−1,−1), where²,η∈ {+,−}. The subcomplexC0 ofC(P0) is de-fined byC0 :=C S−+(+,+), S−+(+,−) +S+−(+,−), S−+(−,+) +S+−(−,−), S−+(−,−) +S+−(−,−)
.
First, the retractionρ:C(P0)→C0 is defined by the formulas S−+(+,+)7→S−+(+,+),
S−+(+,−)7→S−+(+,−) +S+−(+), S−+(−,+)7→S−+(−,+) +S+−(+), S−+(−,−)7→S−+(−,−) +S+−(−), S+−(+,+)7→S−+(+,+),
S+−(−,+)7→S−+(+,−) +S−+(−,+), otherwise7→0.
Second, the isomorphism
C0 →C(P) =C((u| w
∅²| t
∅η|v)) is defined by the formulas
S−+(+,+)7→(u| w
∅+| t
∅+|v), S−+(+,−) +S+−(+)7→(u|
w
∅+| t
∅−|v), S−+(−,+) +S+−(+)7→(u|
w
∅−| t
∅+|v), S−+(−,−) +S+−(−)7→(u|
w
∅−| t
∅−|v).
Third, consider the following composition of this isomorphism with ρ: C(P0)→ρ C0isom→ C(P).
The maph:C(P0)→C(P0) such thatd◦h+h◦d= id−in◦ρ, is defined by the formulas
S−−(²)7→S+−(²,−), S+−(²,+)7→S++(²),
otherwise7→0.
(II–2) Consider the case where the state ofP0 with (mark(A), mark(B)) = (1, 1) is represented as (u|
Aw
∅²| ABt
∅η|v).
S+−(², ζ, η) denotes the state (u| Aw
∅²| AB
∅ζ | Bt
∅η|v) ofP0with (mark(A), mark(B))
= (1,−1),S−+(²) denotes the state (u| ABwt0
∅² |v) ofP0with (mark(A), mark(B))
= (−1, 1), S++(², η) denotes the state (u| Aw
∅²| ABt
∅η |v) of P0 with (mark(A), mark(B)) = (1, 1), and S−−(², η) denotes the state (u|
ABw
∅² | Bt
∅η|v) ofP0 with (mark(A), mark(B)) = (−1, −1), where ², η ∈ {+,−} and the word t0 is ob-tained by deleting all letters from t that appear inw . The subcomplex C0 of C(P0) is defined byC0:=C S−+(+) +S+−(+,−,+),S−+(−) +S+−(+,−,−) +S+−(−,−,+)
.
First, the retractionρ:C(P0)→C0 is defined by the formulas S−+(+)7→S−+(+) +S+−(+,−,+),
S−+(−)7→S−+(−) +S+−(+,−,−) +S+−(−,−,+), S+−(+,+,−)7→S−+(+) +S+−(+,−,+),
S+−(−,+,+)7→S−+(+) +S+−(+,−,+),
S+−(−,+,−)7→S−+(−) +S+−(+,−,−) +S+−(−,−,+), otherwise7→0.
Second, the isomorphism
C0→C(P) =C((u| wt0
∅² |v)) is defined by the formulas
S−+(+) +S+−(+,−,+)7→(u| wt0
∅+|v),
S−+(−) +S+−(+,−,−) +S+−(−,−,+)7→(u| wt0
∅−|v).
Third, consider the following composition of this isomorphism with ρ: C(P0)→ρ C0isom→ C(P).
The maph:C(P0)→C(P0) such thatd◦h+h◦d= id−in◦ρ, is defined by the formulas
S−−(², η)7→S+−(²,−, η), S+−(²,+, η)7→S++(², η),
otherwise7→0.
By using (II-1) and (II-2), we proved thatKHi,j((xAByBAz))'KHi,j((xyz )) if (|A|,|B|) = (1,−1). In addition, (II-1) and (II-2) prove that KHi,j((
xAByABz)) ' KHi,j((xyz)) if (|A|,|B|) = (−1,1). Moreover, by exchang-ingA,B in the proofs above, (II-1) and (II-2) prove thatKHi,j((xAByBAz))' KHi,j((xyz)) if (|A|,|B|) = (−1,1) and KHi,j((xAByABz))'KHi,j((xyz)) if (|A|,|B|) = (1,−1).
Here, consider
xAAyH1∼xABBAy with|A|=−1,|B|= 1
H2∼xy.
We have already shown the invariance of KHi,j under the above moves and that KHi,j is preserved under the first homotopy move xAAy → xy with |A|
=−1 and its inverse move.
(III) Consider the third homotopy move
P0= (xAByACzBCt)→(xBAyCAzCBt) =P
and its inverse move where (|A|,|B|,|C|) = (−1,−1,−1). For the letters A,B, and C, we define wABC, wAB, wAC, wBC, wA, wB, and wC in the following.
Let wABC be a word containing A, B, and C. Let (X, Y, Z) = {(A, B, C), (A, C, B), (B, C, A)}. wXY denotes a word containingX andY but notZ, and wZ denotes a word containingZ but notX andY.
(III–1) Consider the case where the state of P0 with (mark(A), mark(B), mark(C )) = (1, 1, 1) is represented as (u|
wABC
∅² |v).
S+++(²) denotes the state (u| wABC
∅² |v) ofP0with (mark(A), mark(B), mark(
C)) = (1, 1, 1), S−++(², η) denotes the state (u| wABC
∅² | wAB
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (−1, 1, 1), S+−+(², η) denotes the state (u|
wABC
∅² | ABC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (1, −1, 1), S−−+(²) denotes the state (u|
wABC
∅² |v) ofP0with (mark(A), mark(B), mark(C))
= (−1,−1, 1),S++−(², η) denotes the state (u| wABC
∅² | wBC
∅η |v) ofP0with (mark(A), mark(B), mark(C)) = (1, 1,−1), S−+−(², ζ, η) denotes the state (u|
wAC
∅² | wAB
∅ζ
| wBC
∅η |v) ofP0 with (mark(A), mark(B), mark(C)) = (−1, 1,−1),S+−−(²) de-notes the state (u|
wABC
∅² |v) ofP0with (mark(A), mark(B), mark(C)) = (1,−1,
−1), and S−−−(², η) denotes the state (u| wAC
∅² | wABC
∅η |v) ofP0 with (mark(A), mark(B), mark(C)) = (−1,−1,−1).
The subcomplexC0ofC(P0) is defined byC0:=C S−++(+,+),S−++(+,−) +S+−+(+,−),S−++(−,+) +S+−+(+,−),S−++(−,−) +S+−+(−,−),S∗∗−
, where S∗∗−denotes every state with mark(C) =−1.
T+−+(², η) denotes the state (u| wABC
∅² | wC
∅η|v) ofP with (mark(A), mark(B), mark(C)) = (1,−1, 1),T−++(², ζ, η,−) denotes the state (u|
wA
∅²| wB
∅ζ| wC
∅η| ABC
∅− |v) of P with (mark(A), mark(B), mark(C)) = (−1, 1, 1), T++−(², η) denotes the state (u|
wABC
∅² | wB
∅η|v) of P with (mark(A), mark(B), mark(C)) = (1, 1,
−1), T−+−(², ζ, η) denotes the state (u| wA
∅²| wABC
∅ζ | wB
∅η|v) of P with (mark(A), mark(B), mark(C)) = (−1, 1, −1), T+−−(²) denotes the state (u|
wABC
∅² |v) of P with (mark(A), mark(B), mark(C)) = (1,−1,−1), T−−−(², η) denotes the state (u|
wA
∅²| wABC
∅η |v) ofP with (mark(A), mark(B), mark(C)) = (−1,−1,−1), andT∗∗− denotes every state ofP with mark(C) =−1.
The subcomplexCofC(P) is defined byC:=C T+−+(+, η) +T−++(+,+, η,
−),T+−+(−, η) + T−++(+,−, η,−) +T−++(−,+, η,−),T∗∗−
. First, the retractionρ:C(P0)→C0 is defined by the formulas S−++(+,+)7→S−++(+,+),
S−++(+,−)7→S−++(+,−) +S+−+(+,−), S−++(−,+)7→S−++(−,+) +S+−+(+,−), S−++(−,−)7→S−++(−,−) +S+−+(−,−),
S∗∗−7→S∗∗−,
S+−+(+,+)7→S−++(+,+) +S++−(+,+),
S+−+(−,+)7→S−++(+,−) +S−++(−,+) +S++−(+,−) +S++−(−,+), S−−+(²)7→S+−−(²),
otherwise7→0.
Second, consider the following composition of the following isomorphism withρ
C(P0)→ρ C0 isom→ C→i C(P). (4.20)
The isomorphismC0 →Cis defined by the formulas
S−++(+,+)7→T+−+(+,+) +T−++(+,+,+,−), S−++(+,−) +S+−+(+,−)7→T+−+(+,−) +T−++(+,+,−,−), S−++(−,+) +S+−+(+,−)7→T+−+(−,+) +T−++(+,−,+,−)
+T−++(−,+,+,−), S−++(−,−) +S+−+(−,−)7→T+−+(−,−) +T−++(+,−,−,−)
+T−++(−,+,−,−), S++−(², η)7→T++−(², η),
S−+−(², ζ, η)7→T+−+(², ζ, η), S+−−(²)7→T+−−(²), S−−−(², η)7→T−−−(², η).
Third, the map h: C(P0)→ C(P0) such that d◦h+ h◦d = id−in◦ρis defined by the formulas
S−−+(²)7→S+−+(²,−), S+−+(²,+)7→S+++(²),
otherwise7→0.
(III–2) Consider the case where the state of P0 with (mark(A), mark(B), mark(C )) = (1, 1, 1) is represented as (u|
wAC
∅² | wABC
∅η |v).
S+++(², η) denotes the state (u| wAC
∅² | wABC
∅η |v) ofP0 with (mark(A), mark(B ), mark(C)) = (1, 1, 1), S−++(²) denotes the state (u|
wABC
∅² |v) of P0 with (mark(A), mark(B), mark(C)) = (−1, 1, 1), S+−+(², ζ, η) denotes the state (u|
wAC
∅² | ABC
∅ζ | wB
∅η|v) of P0 with (mark(A), mark(B), mark(C)) = (1, −1, 1), S−−+(², η) denotes the state (u|
wABC
∅² | wB
∅η |v) ofP0 with (mark(A), mark(B), mark(C)) = (−1, −1, 1), S++−(²) denotes the state (u|
wABC
∅² |v) of P0 with (mark(A), mark(B), mark(C)) = (1, 1, −1), S−+−(², η) denotes the state (u|
wABC
∅² | wAC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (−1, 1, −1), S+−−(², η) denotes the state (u|
wABC
∅² | wB
∅η|v) of P0 with (mark(A), mark(B), mark(C)) = (1,−1,−1), andS−−−(², ζ, η) denotes the state (u|
wABC
∅² | wAC
∅ζ | wB
∅η| v) ofP0 with (mark(A), mark(B), mark(C)) = (−1,−1,−1).
The subcomplexC0ofC(P0) is defined byC0:=C S−++(+) +S+−+(+,−, +),S−++(−) +S+−+(+,−,−) +S+−+(−,−,+),S∗∗−
, whereS∗∗− denotes every state with mark(C) =−1.
T+−+(²) denotes the state (u| wABC
∅² |v) ofP with (mark(A), mark(B), mark(
C)) = (1, −1, 1), T−++(², ζ,−) denotes the state (u| wBC
∅² | wA
∅ζ| ABC
∅− |v) of P with (mark(A), mark(B), mark(C)) = (−1, 1, 1), T++−(²) denotes the state (u|
wABC
∅² |v) of P with (mark(A), mark(B), mark(C)) = (1, 1,−1), T−+−(², η) denotes the state (u|
wBC
∅² | wA
∅η|v) of P with (mark(A), mark(B), mark(C)) = (−1, 1,−1),T+−−(², η) denotes the state (u|
wABC
∅² | wBC
∅η |v) ofP with (mark(A), mark(B), mark(C)) = (1,−1,−1),T−−−(², ζ, η) denotes the state (u|
wABC
∅² | wA
∅ζ
| wBC
∅η |v) of P with (mark(A), mark(B), mark(C)) = (−1, −1, −1), andT∗∗−
denotes every state ofP with mark(C) =−1.
The subcomplexCofC(P) is defined byC:=C T+−+(+) +T−++(+,+,−), T+−+(−) + T−++(+,−,−) +T−++(−,+,−),T∗∗−
.
First, the retractionρ:C(P0)→C0 is defined by the formulas S−++(+)7→S−++(+) +S+−+(+,−,+),
S−++(−)7→S−++(−) +S+−+(+,−,−) +S+−+(−,−,+), S∗∗−7→S∗∗−,
S+−+(+,+,−)7→S−++(+) +S+−+(+,−,+) +S++−(+), S+−+(−,+,+)7→S−++(+) +S+−+(+,−,+) +S++−(+),
S+−+(−,+,−)7→S−++(−) +S+−+(+,−,−) +S+−+(−,−,−) +S++−(−), S−−+(², η)7→S+−−(², η),
otherwise7→0.
Second, consider the following composition (4.20) of the following isomor-phism withρ. The isomorphismC0 →C is defined by the formulas
S−++(+) +S+−+(+,−,+)7→T+−+(+) +T−++(+,+,−), S−++(−) +S+−+(+,−,−) +S+−+(−,−,+)7→T+−+(−) +T−++(+,−,−)
+T−++(−,+,−), S++−(²)7→T++−(²),
S−+−(², η)7→T−+−(², η), S+−−(², η)7→T+−−(², η), S−−−(², ζ, η)7→T−−−(², ζ, η).
Third, the map h: C(P0)→ C(P0) such that d◦h+ h◦d = id−in◦ρis defined by the formulas
S−−+(², η)7→S+−+(²,−, η), S+−+(²,+, η)7→S+++(², η),
otherwise7→0.
(III–3) Consider the case where the state of P0 with (mark(A), mark(B), mark(C)) = (1, 1, 1) is represented as (u|
wA
∅²| wABC
∅η |v).
S+++(², η) denotes the state (u| wA
∅²| wABC
∅η |v) ofP0with (mark(A), mark(B), mark(C)) = (1, 1, 1),S−++(²) denotes the state (u|
wABC
∅² |v) ofP0with (mark(A), mark(B), mark(C)) = (−1, 1, 1),S+−+(², ζ, η) denotes the state (u|
wA
∅²| ABC
∅ζ
| wBC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (1, −1, 1), S−−+(², η) denotes the state (u|
wABC
∅² | wBC
∅η |v) ofP0 with (mark(A), mark(B), mark(C)) = (−1, −1, 1), S++−(², ζ, η) denotes the state (u|
wA
∅²| wBC
∅ζ | wABC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (1, 1, −1), S−+−(², η) denotes the state (u|
wABC
∅² | wBC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (−1, 1, −1), S+−−(², η) denotes the state (u|
wA
∅²| wABC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (1,−1,−1), andS−−−(²) denotes the state (u|
wABC
∅² |v) ofP0with (mark(A), mark(B), mark(C)) = (−1,−1,−1).
The subcomplexC0ofC(P0) is defined byC0:=C S−++(+) +S+−+(+,−, +),S−++(−) +S+−+(+,−,−) +S+−+(−,−,+),S∗∗−
, whereS∗∗− denotes every states with mark(C) =−1.
T+−+(²) denotes the state (u| wABC
∅² |v) ofP with (mark(A), mark(B), mark(
C)) = (1, −1, 1), T−++(²,−, η) denotes the state (u| wABC
∅² | ABC
∅− | wB
∅η|v) of P with (mark(A), mark(B), mark(C)) = (−1, 1, 1), T++−(², ζ, η) denotes the state (u|
wAC
∅² | wB
∅ζ| wABC
∅η |v) of P with (mark(A), mark(B), mark(C)) = (1, 1,
−1),T−+−(², η) denotes the state (u| wABC
∅² | wB
∅η|v) ofPwith (mark(A), mark(B), mark(C)) = (−1, 1,−1), T+−−(², η) denotes the state (u|
wAC
∅² | wABC
∅η |v) of P with (mark(A), mark(B), mark(C)) = (1,−1,−1), T−−−(²) denotes the state (u|
wABC
∅² |v) ofP with (mark(A), mark(B), mark(C)) = (−1,−1,−1), andT∗∗−
denotes every state ofP with mark(C) =−1.
The subcomplexCofC(P) is defined byC:=C T+−+(+) +T−++(+,−,+), T+−+(−) + T−++(+,−,−) +T−++(−,−,+),T∗∗−
.
First, the retractionρ:C(P0)→C0 is defined by the formulas S−++(+)7→S−++(+) +S+−+(+,−,+),
S−++(−)7→S−++(−) +S+−+(+,−,−) +S+−+(−,−,+), S∗∗−7→S∗∗−,
S+−+(+,+,+)7→S++−(+,+,+),
S+−+(+,+,−)7→S−++(+) +S+−+(+,−,+) +S++−(+,+,−) +S++−(+,−,+),
S+−+(−,+,+)7→S−++(+) +S+−+(+,−,+) +S++−(−,+,+), S+−+(−,+,−)7→S−++(−) +S+−+(+,−,−) +S+−+(−,−,+)
+S++−(−,+,−) +S++−(−,−,+), S−−+(², η)7→S+−−(², η),
otherwise7→0.
Second, consider the following composition (4.20) of the following isomor-phism withρ. The isomorphismC0 →C is defined by the formulas
S−++(+) +S+−+(+,−,+)7→T+−+(+) +T−++(+,−,+), S−++(−) +S+−+(+,−,−) +S+−+(−,−,+)7→T+−+(−) +T−++(+,−,−)
+T−++(−,−,+), S++−(²)7→T++−(²),
S−+−(², η)7→T−+−(², η), S+−−(², η)7→T+−−(², η), S−−−(², ζ, η)7→T−−−(², ζ, η).
Third, the map h: C(P0)→ C(P0) such that d◦h+ h◦d = id−in◦ρis defined by the formulas
S−−+(², η)7→S+−+(²,−, η), S+−+(²,+, η)7→S+++(², η),
otherwise7→0.
(III–4) Consider the case where the state of P0 with (mark(A), mark(B), mark(C)) = (1, 1, 1) is represented as (u|
wABC
∅² | wC
∅η|v).
S+++(², η) denotes the state (u| wABC
∅² | wC
∅η|v) ofP0with (mark(A), mark(B), mark(C)) = (1, 1, 1), S−++(², ζ, η) denotes the state (u|
wABC
∅² | wAB
∅ζ | wC
∅η|v) of P0 with (mark(A), mark(B), mark(C)) = (−1, 1, 1),S+−+(², ζ, η) denotes the state (u|
wAB
∅² | ABC
∅ζ | wC
∅η|v) of P0 with (mark(A), mark(B), mark(C)) = (1, −1, 1),S−−+(², η) denotes the state (u|
wAB
∅² | wC
∅η|v) ofP0 with (mark(A), mark(B),
mark(C)) = (−1, −1, 1), S++−(²) denotes the state (u| wABC
∅² |v) of P0 with (mark(A), mark(B), mark(C)) = (1, 1, −1), S−+−(², η) denotes the state (u|
wABC
∅² | wAB
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (−1, 1, −1), S+−−(², η) denotes the state (u|
wAB
∅² | wABC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (1,−1,−1), andS−−−(²) denotes the state (u|
wABC
∅² |v) ofP0with (mark(A), mark(B), mark(C)) = (−1,−1,−1).
The subcomplexC0ofC(P0) is defined byC0:=C S−++(+,+, η),S−++(+,
−, η) +S+−+(+,−, η),S−++(−,+, η) +S+−+(+,−, η),S−++(−,−, η) +S+−+
(−,−, η),S∗∗−
, where S∗∗−denotes every state with mark(C) = −1.
T+−+(², ζ, η) denotes the state (u| wABC
∅² | wAB
∅ζ | wC
∅η |v) ofP with (mark(A), mark(B), mark(C)) = (1,−1, 1),T−++(²,−, η) denotes the state (u|
wAB
∅² | ABC
∅− | wC
∅η|v) ofPwith (mark(A), mark(B), mark(C)) = (−1, 1, 1),T++−(²) denotes the state (u|
wABC
∅² |v) of P with (mark(A), mark(B), mark(C)) = (1, 1, −1), T−+−(², η) denotes the state (u|
wAB
∅² | wABC
∅η |v) ofP with (mark(A), mark(B), mark(C)) = (−1, 1, −1), T+−−(², η) denotes the state (u|
wABC
∅² | wAB
∅η |v) of P with (mark(A), mark(B), mark(C)) = (1,−1,−1), T−−−(²) denotes the state (u|
wABC
∅² |v) ofP with (mark(A), mark(B), mark(C)) = (−1,−1,−1), andT∗∗−
denotes every state ofP with mark(C) =−1.
The subcomplexCofC(P) is defined byC:=C T+−+(+,+, η),T+−+(+,−, η) +T−++(+,−, η),T+−+(−,+, η) +T−++(+,−, η),T+−+(−,−, η) +T−++(−,
−, η),T∗∗−
.
First, the retractionρ:C(P0)→C0 is defined by the formulas S−++(+,+, η)7→S−++(+,+, η),
S−++(+,−, η)7→S−++(+,−, η) +S+−+(+,−, η), S−++(−,+, η)7→S−++(−,+, η) +S+−+(+,−, η), S−++(−,−, η)7→S−++(−,−, η) +S+−+(−,−, η),
S∗∗−7→S∗∗−,
S+−+(+,+,+)7→S−++(+,+,+),
S+−+(+,+,−)7→S−++(+,+,−) +S++−(+),
S+−+(−,+,+)7→S−++(+,−,+) +S−++(−,+,+) +S++−(+), S+−+(−,+,−)7→S−++(+,−,−) +S−++(−,+,−) +S++−(−),
S−−+(², η)7→S+−−(², η), otherwise7→0.
Second, consider the following composition (4.20) of the following isomor-phism withρ. The isomorphismC0 →C is defined by the formulas
S−++(+,+, η)7→T+−+(+,+, η),
S−++(+,−, η) +S+−+(+,−, η)7→T+−+(+,−, η) +T−++(+,−, η), S−++(−,+, η) +S+−+(+,−, η)7→T+−+(−,+, η) +T−++(+,−, η), S−++(−,−, η) +S+−+(−,−, η)7→T+−+(−,−, η) +T−++(−,−, η),
S++−(²)7→T++−(²), S−+−(², η)7→T−+−(², η), S+−−(², η)7→T+−−(², η),
S−−−(²)7→T−−−(²).
Third, the map h: C(P0)→ C(P0) such that d◦h+ h◦d = id−in◦ρis defined by the formulas
S−−+(², η)7→S+−+(²,−, η), S+−+(²,+, η)7→S+++(², η),
otherwise7→0.
(III–5) Consider the case where the state of P0 with (mark(A), mark(B), mark(C)) = (1, 1, 1) is represented as (u|
wA
∅² | wABC
∅ζ | wC
∅η |v).
S+++(², ζ, η) denotes the state (u| wA
∅²| wABC
∅ζ | wC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (1, 1, 1), S−++(², η) denotes the state (u|
wABC
∅² | wC
∅²|v) of P0 with (mark(A), mark(B), mark(C)) = (−1, 1, 1), S+−+(², ζ, η, θ) de-notes the state (u|
wA
∅²| wB
∅ζ| wC
∅η| ABC
∅θ |v) ofP0with (mark(A), mark(B), mark(C))
= (1, −1, 1), S−−+(², ζ, η) denotes the state (u| wABC
∅² | wB
∅ζ| wC
∅η|v) of P0 with (mark(A), mark(B), mark(C)) = (−1, −1, 1), S++−(², η) denotes the state (u|
wA
∅²| wABC
∅η |v) ofP0with (mark(A), mark(B), mark(C)) = (1, 1,−1),S−+−(²) denotes the state (u|
wABC
∅² |v) of P0 with (mark(A), mark(B), mark(C)) = (−1, 1, −1), S+−−(², ζ, η) denotes the state (u|
wA
∅²| wB
∅ζ| wABC
∅η |v) of P0 with (mark(A), mark(B), mark(C)) = (1,−1,−1), andS−−−(², η) denotes the state (u|
wABC
∅² | wB
∅η|v) ofP0 with (mark(A), mark(B), mark(C)) = (−1,−1,−1).
The subcomplexC0ofC(P0) is defined byC0:=C S−++(+, η) +S+−+(+,+, η,−),S−++(−, η) + S+−+(+,−, η,−) +S+−+(−,+, η,−),S∗∗−
whereS∗∗−
denotes every states with mark(C) =−1.
T+−+(², η) denotes the state (u| wABC
∅² | wAB
∅η |v) ofPwith (mark(A), mark(B), mark(C)) = (1, −1, 1), T−++(²,−) denotes the state (u|
wABC
∅² | ABC
∅− |v) of
P with (mark(A), mark(B), mark(C)) = (−1, 1, 1), T++−(², η) denotes the state (u|
wAC
∅² | wABC
∅η |v) of P with (mark(A), mark(B), mark(C)) = (1, 1,−1), T−+−(²) denotes the state (u|
wABC
∅² |v) ofP with (mark(A), mark(B), mark(C))
= (−1, 1, −1), T+−−(², ζ, η) denotes the state (u| wAC
∅² | wBC
∅ζ | wAB
∅η |v) of P with (mark(A), mark(B), mark(C)) = (1, −1, −1), T−−−(², η) denotes the state (u|
wABC
∅² | wBC
∅η |v) of P with (mark(A), mark(B), mark(C)) = (−1, −1, −1), T∗∗− denotes every state ofP with mark(C) =−1.
The subcomplexCofC(P) is defined byC :=C T+−+(+,+),T+−+(+,−) +T−++(+,−),T+−+(−,+) +T−++(+,−),T+−+(−,−) +T−++(−,−),T∗∗−
. First, the retractionρ:C(P0)→C0 is defined by the formulas
S−++(+, η)7→S−++(+, η) +S+−+(+,+, η,−), S−++(−, η)7→S−++(+,−, η,−) +S+−+(−,+, η,−),
S∗∗−7→S∗∗−, S+−+(+,+,−,+)7→S++−(+,+),
S+−+(+,−,+,+)7→S−++(+,+) +S+−+(+,+,+,−) +S++−(+,+), S+−+(+,−,−,+)7→S−++(+,−) +S+−+(+,+,−,−) +S++−(+,−), S+−+(−,+,+,+)7→S−++(+,+) +S+−+(+,+,+,−),
S+−+(−,+,−,+)7→S−++(+,−) +S+−+(+,+,−,−) +S++−(−,+), S+−+(−,−,+,+)7→S−++(−,+) +S+−+(+,−,+,−) +S+−+(−,+,+,−)
+S++−(−,+), S+−+(−,−,−,+)7→S−++(−,−) +S+−+(+,−,−,−) +S+−+(−,+,−,−)
+S++−(−,−), S−−+(², ζ, η)7→S+−−(², ζ, η),
otherwise7→0.
Second, consider the following composition (4.20) of the following
isomor-phism withρ. The isomorphismC0 →C is defined by the formulas S−++(+,+) +S+−+(+,+,+,−)7→T+−+(+,+), S−++(+,−) +S+−+(+,+,−,−)7→T+−+(+,−)
+T−++(+,−), S−++(−,+) +S+−+(+,−,+,−) +S+−+(−,+,+,−)7→T+−+(−,+)
+T−++(+,−), S−++(−,−) +S+−+(+,−,−,−) +S+−+(−,+,−,−)7→T+−+(−,−)
+T−++(−,−), S++−(², η)7→T++−(², η),
S−+−(²)7→T−+−(²), S+−−(², ζ, η)7→T+−−(², ζ, η),
S−−−(², η)7→T−−−(², η).
Third, the map h: C(P0)→ C(P0) such that d◦h+ h◦d = id−in◦ρis defined by the formulas
S−−+(², ζ, η)7→S+−+(², ζ, η,−), S+−+(², ζ, η,+)7→S+++(², ζ, η),
otherwise7→0.
(III–1) – (III–5) prove thatKHi,j((xAByACzBCt))'KHi,j((xBAyCAzCB t)) if (|A|,|B|,|C|) is any among {(−1,−1,−1), (−1,1,1), (1,1,−1)}.
Consider P0 = (xAByACzBCt)→(xBAyCAzCBt) = P, where (|A|,|B|,
|C|) = (1,−1,−1).
xAByACzBCtν−∼shiftxBCyABzACt with (|A|,|B|,|C|) = (1,−1,−1)
isom' xAByDAzDBt with (|A|,|B|,|D|) = (−1,−1,1)
Lemma 4
∼ xAByDACEzDBCEt with|C|=−1,|E|= 1
H3∼ xBAyDCAEzDCBEt with (|A|,|B|,|C|) = (−1,−1,−1)
H2∼ xBAyAEzBEt with|C|=−1,|D|= 1
isom' xCByBAzCAt with (|A|,|B|,|C|) = (1,−1,−1)
ν−shift
∼ xBAyCAzCBt with (|A|,|B|,|C|) = (1,−1,−1) We have already shown the invariance of KHi,j under the above moves and that KHi,j is preserved under the third homotopy move H3 and its inverse move with (|A|,|B|,|C|) = (1,−1,−1). In particular, in this case, we use the invariance ofKHi,j under H3 and its inverse with (|A|, |B|, |C|) = (−1, −1,
−1). By using the invariance under H3 and its inverse with (|A|,|B|,|C|) = (1,
1, −1) (resp. (−1, 1, 1)), we can verify the invariance ofKHi,j under H3 and its inverse with (|A|, |B|,|C|) = (1,1,1) (resp. (−1,−1,1)).
We conclude that KHi,j(P0)'KHi,j(P) forP0 'S1 P. The following corollary is a similar to Corollary 14.
Corollary 15. KHi,j(P) are S0-homotopy invariants for nanophrases P over α0.