Chirally cosmetic banding on knots and links

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1 Chirally cosmetic banding on knots and links In Dae Jong Kindai University joint work with Kazuhiro Ichihara (Nihon University) The 8th KOOK-TAPU National University 2016/7/28 11:00 11:45 1 / 27

2 Contents 1. Introduction Definitions on cosmetic banding Questions 2, 3. Several examples on cosmetic bandings 4. Banding and Dehn filling, Dehn surgery 5. Bleiler-Hodgson-Weeks s example and a conjecture 6. Main Result (which gives a counterexample of a conjecture) 2 / 27

3 1. Introduction - banding - L : a link in S 3, b: [0, 1] [0, 1] S 3 : embedding s.t. b([0, 1] [0, 1]) L = b([0, 1] {0, 1}) Definition (banding) (also called band surgery) L = (L b(i I)) b( I I) is called the link obtained from L by a banding along the band b. banding L L 3 / 27

4 1. Introduction - banding - L : a link in S 3, b: [0, 1] [0, 1] S 3 : embedding s.t. b([0, 1] [0, 1]) L = b([0, 1] {0, 1}) Definition (banding) (also called band surgery) L = (L b(i I)) b( I I) is called the link obtained from L by a banding along the band b. banding Question L L For a link L, does a banding give the same link? (i.e. L L?) 3 / 27

5 1. Introduction - cosmetic banding - L : a link in S 3, L : a link obtained from L by a banding Definition (cosmetic banding) A banding is cosmetic def h: S 3 S 3 : homeo. s.t. h(l) = L. Cosmetic = leaving their types unchanged. 4 / 27

6 1. Introduction - cosmetic banding - L : a link in S 3, L : a link obtained from L by a banding Definition (cosmetic banding) A banding is cosmetic def h: S 3 S 3 : homeo. s.t. h(l) = L. Cosmetic = leaving their types unchanged. Example: cosmetic non-cosmetic 4 / 27

7 1. Introduction - pure/chiral - L S 3, L : a link obtained from L by a cosmetic banding h: S 3 S 3 : homeo. s.t. h(l) = L. Definition (purely / chirally cosmetic) A banding is purely (resp. chirally) cosmetic def h is orientation preserving (resp. reversing). 5 / 27

8 1. Introduction - pure/chiral - L S 3, L : a link obtained from L by a cosmetic banding h: S 3 S 3 : homeo. s.t. h(l) = L. Definition (purely / chirally cosmetic) A banding is purely (resp. chirally) cosmetic def h is orientation preserving (resp. reversing). h is orientation preserving L L in S 3. h is orientation reversing L the mirror image of L in S 3. 5 / 27

9 1. Introduction - pure/chiral - L S 3, L : a link obtained from L by a cosmetic banding h: S 3 S 3 : homeo. s.t. h(l) = L. Definition (purely / chirally cosmetic) A banding is purely (resp. chirally) cosmetic def h is orientation preserving (resp. reversing). h is orientation preserving L L in S 3. h is orientation reversing L the mirror image of L in S 3. This banding is purely and also chirally cosmetic. (trivial example) 5 / 27

10 1. Introduction - questions - Find various (purely or chirally) cosmetic bandings. For a given cosmetic banding, reveal the reason why the banding is cosmetic. Today we will consider them. 6 / 27

11 1. Introduction - questions - Find various (purely or chirally) cosmetic bandings. For a given cosmetic banding, reveal the reason why the banding is cosmetic. Today we will consider them. Study related problems cosmetic surgery conjecture generalized knot complement conjecture etc... 6 / 27

12 2. Known examples - banding and 4-move - 4-move equivalent to 7 / 27

13 2. Known examples - banding and 4-move - 4-move Lemma A banding yields a 4-move. equivalent to 7 / 27

14 2. Known examples - pretzel knots - (n, 2, n)-pretzel knot P (n, 2, n) P (n, 2, n) π-rot. mirror image P ( n, 2, n) 8 / 27

15 2. Known examples - Kinoshita-Terasaka knot - Kinoshita-Terasaka knot banding vertical π-rot. mirror image 9 / 27

16 2. Known examples - Kinoshita-Terasaka knot - Kinoshita-Terasaka knot banding vertical π-rot. mirror image More generally, particular symmetric unions 9 / 27

17 2. Known examples - two-bridge links with u(l) = 1 - S(2m 2, 2mn 1) = C(a 0, a 1,..., a k, 2, a k,..., a 1, a 0 ) (2-bridge links with unlinking number 1 [Kohn]) banding horizontal π-rot. mirror image 10 / 27

18 2. Known examples - two-bridge links with u(l) = 1 - S(2m 2, 2mn 1) = C(a 0, a 1,..., a k, 2, a k,..., a 1, a 0 ) (2-bridge links with unlinking number 1 [Kohn]) banding horizontal π-rot. mirror image...these examples are trivial in some sense / 27

19 2. Known examples - mundane/exotic - banding with slope n replacing a 1/0-tangle into an n-tangle. 1/0-tangle 3-tangle banding ball B 11 / 27

20 2. Known examples - mundane/exotic - banding with slope n replacing a 1/0-tangle into an n-tangle. 1/0-tangle 3-tangle banding ball B T B := S 3 \ intb is regarded as a tangle. Definition (mundane/exotic cosmetic banding) A cosmetic banding with slope n is mundane def h: homeo. (as a tangle) on T B sending the slope n to 1/0. exotic def it is not mundane. 11 / 27

21 2. Known examples - 4-move banding is mundane - twisting mirroring orientation reversing homeomorphism sending the slope 1 to 1/0 12 / 27

22 3. Zeković s example - (2,5)-torus knot - Proposition [Zeković 14] The (2, 5)-torus knot admits a cosmetic banding / 27

23 3. Zeković s example - (2,5)-torus knot - Proposition [Zeković 14] The (2, 5)-torus knot admits a cosmetic banding. 0 We will show that this banding cannot be realized by a 4-move, this banding is mundane. 13 / 27

24 3. Zeković s example - can t be realized by a 4-move - σ(k): the signature of a knot K. Proposition [Murasugi 65] K a single crossing change K σ(k) σ(k ) / 27

25 3. Zeković s example - can t be realized by a 4-move - σ(k): the signature of a knot K. Proposition [Murasugi 65] K a single crossing change K σ(k) σ(k ) 2. Since a 4-move is realized by crossing changes two times, K On the other hand, a single 4-move K σ(k) σ(k ) 4. σ(t (2, 5)) = 4, σ(t (2, 5)) = 4. Therefore this chirally cosmetic banding on T (2, 5) cannot be realized by a 4-move. 14 / 27

26 3. Zeković s example - It is mundane - 0-slope banding mirroring To show: They are ambient isotopic with fixing the of the red band. 15 / 27

27 3. Zeković s example - It is mundane - 16 / 27

28 4. Banding and Dehn filling, Dehn surgery X : an oriented compact 3-mfd. with X = T 2 X(r) : the 3-mfd. obtained by r-dehn filling on X (r: slope) Definition (cosmetic filling) r 1 -filling and r 2 -filling on X are purely (resp. chirally) cosmetic def h: X(r 1 ) X(r 2 ): ori. preserving (resp. reversing) homeo. 17 / 27

29 4. Banding and Dehn filling, Dehn surgery X : an oriented compact 3-mfd. with X = T 2 X(r) : the 3-mfd. obtained by r-dehn filling on X (r: slope) Definition (cosmetic filling) r 1 -filling and r 2 -filling on X are purely (resp. chirally) cosmetic def h: X(r 1 ) X(r 2 ): ori. preserving (resp. reversing) homeo. mundane def ϕ: X X: homeo. s.t. ϕ(r 1 ) = r 2 exotic def not mundane 17 / 27

30 4. Banding and Dehn filling, Dehn surgery X : an oriented compact 3-mfd. with X = T 2 X(r) : the 3-mfd. obtained by r-dehn filling on X (r: slope) Definition (cosmetic filling) r 1 -filling and r 2 -filling on X are purely (resp. chirally) cosmetic def h: X(r 1 ) X(r 2 ): ori. preserving (resp. reversing) homeo. mundane def ϕ: X X: homeo. s.t. ϕ(r 1 ) = r 2 exotic def not mundane ex. Purely mundane cosmetic fillings: ex. For amphichiral knot K and r Q, (±r)-fillings on the exterior E(K) are chirally mundane cosmetic. 1 n = / 27

31 4. Banding and Dehn filling, Dehn surgery Cosmetic Surgery Conjecture Cosmetic fillings are never purely and exotic. 18 / 27

32 4. Banding and Dehn filling, Dehn surgery Cosmetic Surgery Conjecture Cosmetic fillings are never purely and exotic. Proposition [Mathieu 92] X = E(the trefoil knot), (k 0) 18k + 9 3k k and -fillings on X are chirally exotic cosmetic. 3k + 2 Remark In Mathieu s example, X and the resultants are Seifert. 18 / 27

33 4. Banding and Dehn filling, Dehn surgery Lemma (called Montesinos trick ) K : a knot in S 3 admitting a cosmetic banding Σ K : the double branched cover of S 3 branched along K We have cosmetic fillings on the exterior of a knot in Σ K. 19 / 27

34 4. Banding and Dehn filling, Dehn surgery Lemma (called Montesinos trick ) K : a knot in S 3 admitting a cosmetic banding Σ K : the double branched cover of S 3 branched along K We have cosmetic fillings on the exterior of a knot in Σ K. 0 double branched cover 5 1 The complement of the knot in L(5, 1) is called the figure eight sibling. It is known that the figure eight sibling is amphichiral. 19 / 27

35 5. Bleiler-Hodgson-Weeks s example Proposition [Bleiler-Hodgson-Weeks] a hyperbolic knot K S 2 S 1 s.t. E(K) admits chirally exotic cosmetic fillings yielding L(49, ±18). (L(49, 18) = L(49, 19)) K Conjecture [Bleiler-Hodgson-Weeks] Cusped hyperbolic manifolds admit no exotic cosmetic fillings, purely or chirally, yielding hyperbolic mfds / 27

36 5. Bleiler-Hodgson-Weeks s example Proposition [Bleiler-Hodgson-Weeks] a hyperbolic knot K S 2 S 1 s.t. E(K) admits chirally exotic cosmetic fillings yielding L(49, ±18). (L(49, 18) = L(49, 19)) K Conjecture [Bleiler-Hodgson-Weeks] Cusped hyperbolic manifolds admit no exotic cosmetic fillings, purely or chirally, yielding hyperbolic mfds. 0 We give a counterexample for the chirally case. 20 / 27

37 6. Result - Statement - Theorem [Ichihara-J.] a 1-cusped hyperbolic mfd. admitting exotic chirally cosmetic fillings which yield hyperbolic manifolds. 3 or / 27

38 6. Main Result - construction - Proposition [Ichihara-J.] 9 27 = S(49, 18) admits a chirally exotic cosmetic banding. 1 1 Remark This is obtained from the cosmetic fillings of [B.-H.-W.] = C(1, 1, 1, 2, 1, 1, 1, 2) = C(2, 2, 2, 2, 2, 2) 9 27 is a symmetric union of 5 2. In particular, it is ribbon. 22 / 27

39 6. Main Result - construction can be obtained as follows: 23 / 27

40 6. Main Result - construction - mirroring & 2π/3-rot. 0-slope banding 24 / 27

41 6. Main Result - construction - mirroring & 2π/3-rot. 0-slope banding This yields many chirally cosmetic banding by adding twists of the tangles, or increasing the number of tangles from 3 2 to n 2, / 27

42 6. Main Result - construction - Proposition [Ichihara-J.] The knot K admits a chirally exotic cosmetic banding. K 25 / 27

43 6. Main Result - construction /2 1 3/ /2 3/ cut at / 27

44 6. Main Result - construction K K Σ K Using computer programs SnapPy and hikmot, it is shown that Σ K and Σ K \ K are hyperbolic. The slopes 3 and 1/0 on N( K) in Σ K are inequivalent. (i.e. these cosmetic fillings are exotic.) 27 / 27

NOTES ON MULTI FUZZY RW-OPEN MAPS, MULTI FUZZY RW-CLOSED MAPS AND MULTI FUZZY RW- HOMEOMORPHISMS IN MULTI FUZZY TOPOLOGICAL SPACE

International Journal of Applied Mathematical Sciences ISSN 0973-0176 Volume 9, Number 1 (2016), pp. 73-87 Research India Publications http://www.ripublication.com NOTES ON MULTI FUZZY RW-OPEN MAPS, MULTI