Knitting pattern with three components
吉野 馨
埼玉大学大学院理工学研究科 修士 2年
2020.12.24
結び目の数理 III
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 1 / 43
Motivation
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 2 / 43
Today’s talk
Definitions of knitting and its pattern
Equivalence of them
Mathematical model of knitting
patterns obtained from model 1
Classification of how green string is
entangled in mathematical model
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 3 / 43
Knitting
R = ⟨v1,v2⟩ : basis of R2∀u = (p, q) ∈ Z2, ∀x ∈ R2, ΦR(u,x) = φu(x) = x+ pv1 + qv2⇒ ΦR : Z2-action.Definition (Knitting [Kawauchi])
K̃ : quadrivalent graph which are embedded in R2 and has heightinformation at each of the vertices(crossing points).
K̃ ⊂ R2 : knitting def⇐⇒ K̃ is preserved by Z2-action ΦR
(i.e., ∃R = ⟨v1,v2⟩: basis of R2 s.t. ∀u ∈ Z2, φu(K̃) = K̃ and φupreserves height information at each of the crossing points).
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 4 / 43
Knitting
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 5 / 43
Equivalence of knitting
Definition
R,R′ : basis of R2, ΦR,ΦR′ : Z2-actionsDiffeomorphism g̃ : (ΦR,ΦR
′)-equivalent
def⇐⇒ R2 g̃ //φu
��⟳
R2
φ′u��
R2g̃
// R2
Definition (Equivalence of knittings)
K̃, K̃ ′ : knitting
K̃ and K̃ ′ are equivalent (K̃ ∼ K̃ ′)def⇐⇒ ∃g̃ : (ΦR,ΦR′)-equivalent diffeo.
s.t. g̃(K̃) ⇝ K̃ ′ : Z2-equivalent Reidemeister moves.Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 6 / 43
Example
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 7 / 43
Knitting pattern
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 8 / 43
Knitting pattern
Definition (Knitting pattern)
Q ⊂ R2 : closed set, intQ : connected, (intQ) = QQ : fundamental region of K̃def⇐⇒ The following conditions are satisfied.1
⋃u=(p,q)∈Z2
φu(Q) = R2
2 φu(intQ) ∩ φu′(intQ) = ∅ (u ̸= u′)We assume Q ∼= I × I.Then we call the tangle diagram K = K̃ ∩Q a knitting pattern.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 9 / 43
Link diagram on T 2
We can obtain a link diagram on T 2 from a knitting pattern.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 10 / 43
Link diagram on T 2
T̂ = T 2 × Ipr : T̂ → T 2 × {0}
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 11 / 43
Equivalence of knitting pattern
Definition (TR-equivalent)
KT , K′T : link diagrams on T
2 obtained from knitting patterns K, K ′.
KT and K′T are TR-equivalent (KT ∼TR K ′T )
def⇐⇒ ∃g : orientation preserving diffeomorphisms.t. g(KT ) ⇝ K ′T : Reidemeister moves on T 2.
Definition
Knitting patterns K and K ′ are equivalent (K ∼ K ′).def⇐⇒ KT ∼TR K ′T .
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 12 / 43
Primitive knitting pattern
Definition
K̃ : knitting, Q : fundamental region of K̃
K : knitting pattern obtained from K̃
K : primitivedef⇐⇒ The area of Q is minimum among all of knitting
patterns of K̃.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 13 / 43
Theorem 1
Theorem 1
K̃, K̃ ′ : knittings
K, K ′ : primitive knitting patterns obtained from K̃ and K̃ ′.
Then, K̃ ∼ K̃ ′ ⇔ K ∼ K ′
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 14 / 43
From now, we consider model 1.
Properties of model 1
Consisting of three components and
these are essential simple loops on
T 2.
A red string passes over a blue
string.
A green string passes through the
complement areas of other strings
once.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 15 / 43
Step 1 Consider the “slope” for an essential simple component
of knitting pattern (Loops in model 1 have slope 12, −1
2
and ∞).
Step 2 Classify link projections on T 2 by “word”.
Step 3 We get link diagrams on T 2 from link projections
obtained in step 2.
In the end, we give the classification of the case n = 2. (Theorem 3)
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 16 / 43
Slope for a component of a knitting pattern
We consider knitting K̃ consisting only of components which are
homeomorphic to R and these components induce essential simpleloops on T 2.
Definition (Slope for a knitting pattern)
l : essential simple loop on T 2
l has slope qp(p ∈ Z≥0, q ∈ Z, gcd(p, q) = 1)
Then, a component l in knitting pattern has slope qp(l = lq/p).
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 17 / 43
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 18 / 43
Mathematical model
Properties of model 1
Consisting of essential
simple loops l1/2, l−1/2 and
l∞ on T2.
l∞ passes through the
complement areas of
l1/2 ∪ l−1/2 once.
G : graph on T 2 dual to
l1/n ∪ l−1/n.lq/p : essential simple loop on G.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 19 / 43
Assumption
n ∈ N, p ∈ Z≥0, q ∈ Z, gcd(p, q)=1We consider knitting pattern with three components l1/n, l−1/n and
lq/p on G.
Definition
p1, p2, q1, q2 ∈ Z, gcd(p1, q1)=gcd(p2, q2)=1
∆(q1/p1, q2/p2)=|p2q1 − p1q2|
denotes the minimum crossing number
between lq1/p1 and lq2/p2 .
Observation
c(l1/n, lq/p) = ∆(1/n, q/p)
(resp. c(l−1/n, lq/p) = ∆(−1/n, q/p))Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 20 / 43
Link projection on T 2
Label the crossing point between lq/p and l1/n as r, and l−1/n as b.
Then, we get a “word” generated by {r, b}.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 21 / 43
Link projection on T 2
Proposition
l1/n, l−1/n : essential s.c.c. on T2 with slope 1
n, − 1
n
lq/p : essential s.c.c. on T2 on G
#r = c(l1/n, lq/p), #b = c(l−1/n, lq/p)
Then,qp= 0 ⇒ c = #r +#b = 1 + 1 = 2
qp̸= 0 ⇒ c = #r +#b = 2n
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 22 / 43
Link projection on T 2
From now, we consider the case qp̸= 0, and
we can assume |q| = 1.
Remark
wq/p : a word induced from lq/p#r = #{r ∈ wq/p}, #b = #{b ∈ wq/p}
Proposition
l1/n, l−1/n : essential s.c.c. on T2 with slope 1
n, − 1
n
lq/p : essential s.c.c. on T2 on G
#r = c(l1/n, lq/p), #b = c(l−1/n, lq/p)
∀t ∈ {0, 1, . . . , 2n}, (#r,#b) = (t, 2n− t) ⇔ qp= 1
n−t
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 23 / 43
Link projection on T 2
In this slide, we consider link projections on T 2
Definition (Equivalence of word)
Two words with length 2n generated by {r, b} are equivalentdef⇐⇒ Two words are same up to circular permutation.
Definition (Equivalence of link projection on T 2)
Two link projections are equivalentdef⇐⇒ ∃orientation preserving diffeomorphism of T 2 taking oneprojection to the other.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 24 / 43
Link projection on T 2
Proposition
w = x1x2 · · ·x2n (xi ∈ {r, b}) : wordw′ = x′1x
′2 · · · x′2n (x′i ∈ {r, b}) : word
KT , K′T : link projections on T
2 induced by w and w′.
Then,
w and w′ are equivalent ⇔ KT and K ′T are equivalent
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 25 / 43
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 26 / 43
Link diagram on T 2
We consider link diagrams on T 2 with three components l1/n, l−1/nand lq/p, and l1/n ≥ l−1/n(i.e. l1/n passes over l−1/n at all of thecrossing points.) holds.
Label the crossing points between lq/p and l1/n∪ l−1/n in the linkdiagram on T 2 either r, r, b, or b.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 27 / 43
Example
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Split link diagram
Definition
D : link diagram on T 2 with n components l1, l2, . . . , ln (n ∈ N)li ≥ lj (resp. li ≤ lj) (i, j ∈ {1, 2, . . . , n}, i ̸= j)def⇐⇒ li passes over (resp. under) lj at all of the crossing points of liand lj or li ∩ lj = ∅ in D.D : splitdef⇐⇒ ∃k ∈ {1, 2, . . . , n} s.t. ∀m ∈ {1, . . . , k − 1, k + 1, . . . , n},lk ≥ lmlk ≤ lm
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 29 / 43
Example
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 30 / 43
Non split link diagram on T 2
Proposition
w = x1x2 · · ·x2n (xi ∈ {r, r, b, b}) : wordKT : link diagram on T
2 induced by w with l1/n ≥ l−1/nxi ∈ {r, b, b}xi ∈ {r, r, b} ⇔ KT : split
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 31 / 43
RIII-equivalent
Definition (RIII-equivalent)
KT , K′T : link diagrams on T
2 obtained from knitting patterns K, K ′.
KT and K′T are RIII-equivalent
def⇐⇒ ∃g : orientation preserving diffeomorphisms.t. g(KT ) ⇝ K ′T : Reidemeister moves III on T 2.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 32 / 43
Reidemeister move III
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 33 / 43
Link diagram on T 2
Definition (Equivalence of word)
Two words with length 2n generated by {r, r, b, b} are equivalentdef⇐⇒ Two words are same words up to circular permutation afterfinite sequence of wRIIIs.
Theorem 2
w = x1x2 · · ·x2n (xi ∈ {r, r, b, b}) : wordw′ = x′1x
′2 · · · x′2n (x′i ∈ {r, r, b, b}) : word
KT , K′T : link diagrams on T
2 induced by w and w′ with l1/n ≥ l−1/n
w and w′ are equivalent ⇔ KT and K ′T are RIII-equivalent
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 34 / 43
Result
Algorithm
n ∈ N, t ∈ {0, 1, . . . , 2n}How to get knitting pattern consisting of l1/n, l−1/n and l1/(n−t).
1 Consider words generated by {r, b} with (#r,#b) = (t, 2n− t)up to cyclic permutation.
2 Give hight information to words and exclude word consisting of
{r, b, b} or {r, r, b}.(Exclude split knitting patterns)3 Consider words up to cyclic permutation after finite sequences of
wRIIIs.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 35 / 43
Theorem 3(Classification of the case n = 2)
p ∈ {0, 1, 2}, |q| = 1, gcd(p, q)=1l1/2, l−1/2, lq/p : essential simple loops on T
2 with slope 12, −1
2and q
p
There are nine non split knitting patterns consisting of l1/2, l−1/2 and
lq/p.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 36 / 43
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 37 / 43
Future works
The relation between equivalence of words and equivalence of
link diagrams on T 2 up to Reidemeister moves I, II and III.
Knitting patterns consisting of ln/m, l−n/m and lq/p.
Considering model 2.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 38 / 43
Reference
A. Kawauchi, Complexities of a knitting pattern, Reactive and
Functional Polymers, 131, (2018) 230- 236.
A. A. Akimova, S. V. Matveev and V. V. Tarkaev, Classification
of Links of Small Complexity in the Thickened Torus,
Proc.SteklovInst.Math. 303, (Suppl 1) (2018) 12- 24.
A. A. Akimova, S. V. Matveev and V. V. Tarkaev, Classification
of prime links of in the thickened torus having crossing number 5,
Journal of Knot Theory and Its Ramifications 29(03), (2020)
2050012.
A. A. Akimova, Classification of knots in a thickened torus whose
minimal octahedral diagrams do not lie in an annulus, Vestn.
Yuzhno-Ural. Gos. Univ., Ser. Mat. Mekh. Fiz. 7(1), (2015)
5- 10.Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 39 / 43
Reference
Sergey V. Matveev, Prime decompositions of knots in T 2 × I,Topology and its Applications, 159, (2012) 1820- 1824.
S. Grishanov, V. Meshkov and A. Omelchenko, A Topological
Study of Textile Structures. Part I: Topological Invariants in
Application to Textile Structures, Textile Research Journal,
79(8), (2009) 702- 713.
S. Grishanov, V. Meshkov and A. Omelchenko, A Topological
Study of Textile Structures. Part II: Topological Invariants in
Application to Textile Structures, Textile Research Journal,
79(9), (2009) 822- 836.
Kaoru Yoshino (Saitama University) Knitting pattern with three components 2020.12.24 結び目の数理 III 40 / 43