strong and weak (1, 2) homotopies on knot …mok7/slide/takimura_s.pdfstrong and weak (1, 2)...
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![Page 1: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/1.jpg)
Strong and weak (1, 2) homotopies on knot projections and new invariants
学習院中等科 瀧村祐介
伊藤昇氏(早稲田大学)との共同研究
![Page 2: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/2.jpg)
Definition
k: knot in R³ p : R³ → R² ⊂ R² U {∞} ≅ S² p(k) : knot projection of k 以後、P と表す
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Definition
Reidemeister moves on knot projection
![Page 4: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/4.jpg)
Proposition
∀P₁ , P₂ : knot projections on S
P₁ , P₂ は RI, RII, RIII で移り合う
2
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![Page 6: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/6.jpg)
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Definition(2013)
![Page 8: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/8.jpg)
Definition(2013)
P を 1b,2b で reduced にしたものを P と表す r
![Page 9: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/9.jpg)
![Page 10: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/10.jpg)
Theorem(2013) P₁ と P₂ が RI, RII で移り合う
P₁ と P₂ が球面上で 同じ knot projection である
Corollary P において、 P は一意的である
r r
r
![Page 11: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/11.jpg)
Eliahou, Harary, Kauffman (2008)
Lune free knot graphs
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Definition
![Page 15: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/15.jpg)
Definition
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Definition
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Definition
P を 1b, strong2b で reduced にしたものを P と表す sr
P を 1b, weak2b で reduced にしたものを P と表す wr
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Theorem P₁ と P₂ が strong(1, 2) で移り合う
P₁ と P₂ が球面上で 同じ knot projection である
Corollary P において、 P は一意的である
sr sr
sr
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Theorem P₁ と P₂ が weak(1, 2) で移り合う
P₁ と P₂ が球面上で 同じ knot projection である
Corollary P において、 P は一意的である
wr wr
wr
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P に向きをつけ、全ての交点において以下の smoothing をしたときに出来るcircle の球面上の配置を τ(P) 、 circle の数を|τ(P)| と表す。
Definition
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P
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P
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P
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τ(P) P
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|τ(P)| = 3
τ(P) P
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Theorem
circle の配置 τ(P) と
circle number |τ(P)| は
strong(1, 2) において不変である
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- strong(1, 2)
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Theorem
circle number |τ(P)| は奇数である
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⇒
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⇒
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⇒
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不変
2変化
⇒
![Page 49: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/49.jpg)
![Page 50: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/50.jpg)
![Page 51: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/51.jpg)
![Page 52: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/52.jpg)
![Page 53: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/53.jpg)
![Page 54: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/54.jpg)
不変
2変化
![Page 55: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/55.jpg)
|τ(P)|
![Page 56: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/56.jpg)
mod (2) で不変
|τ(P)|
![Page 57: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/57.jpg)
mod (2) で不変
|τ(○)| = 1
|τ(P)|
![Page 58: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/58.jpg)
Theorem
circle number |τ(P)| は奇数である
![Page 59: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/59.jpg)
Proposition
|τ(P#P′)| = |τ(P)| + |τ(P′)| – 1
(連結和において、加法性が成り立つ)
![Page 60: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/60.jpg)
![Page 61: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/61.jpg)
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|τ(P#P′)| = |τ(P)| + |τ(P′)| – 1
5= 3 + 3 - 1
![Page 63: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/63.jpg)
Theorem
任意の circle 配置 τ(P) の
knot projection P が存在する
![Page 64: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/64.jpg)
![Page 65: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/65.jpg)
Lemma
任意の τ(P) は、以下のどちらかを部分的に含む
![Page 66: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/66.jpg)
Lemma
![Page 67: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/67.jpg)
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![Page 69: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/69.jpg)
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![Page 73: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/73.jpg)
![Page 74: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/74.jpg)
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![Page 80: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/80.jpg)
![Page 81: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/81.jpg)
Theorem
任意の circle 配置の prime な
knot projection P が 存在する
![Page 82: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/82.jpg)
P において、任意の分解円周が trivial arc であるとき、P を prime であるという。
Definition
prime prime ではない
![Page 83: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/83.jpg)
Lemma
prime ではない knot projection でも
有限回のRI とstrong RII でprime に出来る
![Page 84: Strong and weak (1, 2) homotopies on knot …mok7/slide/Takimura_s.pdfStrong and weak (1, 2) homotopies on knot projections and new invariants 学習院中等科 瀧村祐介 伊藤昇氏(早稲田大学)との共同研究](https://reader033.vdocuments.pub/reader033/viewer/2022041915/5e69724a8a0c8f0cdf0c6fc8/html5/thumbnails/84.jpg)
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Thank you for listening