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Unknotting Sequence of Torus Knots and Unknotting Numbers

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Unknotting Sequence of Torus Knots and Unknotting Numbers Unknotting Sequence of Torus Knots and Unknotting Numbers Vikash Siwach (A joint work with Dr. M. Prabhakar) Indian Institute of Technology Ropar India Knots and Low Dimensional Manifolds BEXCO, Busan, Korea August 22-26, 2014 Objective Minimal Unknotting Crossing Data for Torus Knots Bibliography Preliminaries Unknotting Sequence: An unknotting sequence for a knot or a link K is a finite sequence of knots or links K = Kn, Kn−1, Kn−2, ··· , K1, K0 = trivial link, such that: (1) u(Ki ) = i (2) Ki−1 is obtained from Ki by one crossing change. 2 / 23 Objective Minimal Unknotting Crossing Data for Torus Knots Bibliography Preliminaries Unknotting Sequence: An unknotting sequence for a knot or a link K is a finite sequence of knots or links K = Kn, Kn−1, Kn−2, ··· , K1, K0 = trivial link, such that: (1) u(Ki ) = i (2) Ki−1 is obtained from Ki by one crossing change. Minimal unknotting crossing data (MUCD): Any minimal set of crossings in a diagram D of a knot K whose change transform K in to a trivial knot. 2 / 23 Objective Minimal Unknotting Crossing Data for Torus Knots Bibliography Preliminaries Unknotting Sequence: An unknotting sequence for a knot or a link K is a finite sequence of knots or links K = Kn, Kn−1, Kn−2, ··· , K1, K0 = trivial link, such that: (1) u(Ki ) = i (2) Ki−1 is obtained from Ki by one crossing change. Minimal unknotting crossing data (MUCD): Any minimal set of crossings in a diagram D of a knot K whose change transform K in to a trivial knot. Different permutations(ordering) of crossing change from MUCD give dif- ferent unknotting sequences. 2 / 23 ( Identification of intermediate knots in Knot table ) Objective Minimal Unknotting Crossing Data for Torus Knots Bibliography Objective In this Presentation, we discuss Minimal unknotting crossing data for Torus Knots Unknotting Number of Knots 3 / 23 Objective Minimal Unknotting Crossing Data for Torus Knots Bibliography Objective In this Presentation, we discuss Minimal unknotting crossing data for Torus Knots Unknotting Number of Knots ( Identification of intermediate knots in Knot table ) 3 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Minimal Unknotting Crossing Data for Torus Knots1 1V. Siwach and P. Madeti, A Method for Unknotting Torus Knots math.GT/1207.4918v1, 2012. 4 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Preliminaries U-Crossing data denoted by U(B(p, q)): When p > q B(8, 6) U(B(8, 6)) = [14, 20, 21, 26, 27, 28, 32, 33, 34, 35, 38, 39, 40, 41, 42] 5 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Preliminaries When p < q B(8, 13) U(B(8, 13)) = [14, 20, 21, 26, 27, 28, 32, 33, 34, 35, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 70, 76, 77, 82, 83, 84, 88, 89, 90, 91] 6 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results Theorem 1: Let K(p, q) be a torus knot with (p, q) = 1. If q ≡ 1 or p − 1 (mod p), then the U−crossing data for B(p, q) is a minimal unknotting crossing data for B(p, q) (or K(p, q)). 7 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Example: when q ≡ 1 (mod p) Unknotting procedure for K(5, 6). K(5,6) unknot 8 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Example: when q ≡ p − 1 (mod p) Unknotting procedure for K(5, 4). K(5,4) unknot 9 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results Theorem 2: For every p−braid, where p > a for some a, we have −1 −1 −1 η1σp−aσp−a+1 ··· σp−1 η2σp−aσp−a+1 ··· σp−1 ··· ηaσp−aσp−a+1 ··· σp−1 | {z } | {z } | {z } 1 2 a ∼M η1η2 ··· ηa ei,1 ei,2 ei,n−1 where, ηi = σ1 σ2 ··· σp−a−1 (1 ≤ i ≤ a), ei,j is either 1 or −1. 10 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 11 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Example Unknotting procedure for B(7, 4) 12 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Example Minimal Unknotting Crossing Data for B(7, 4) 13 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Unknotting number of some knots 14 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography MUCD for K(5,6) If we change {8, 11, 12, 16, 18, 20} crossings in K(5, 6), we get n1414274. So u(n1414274) = 4 15 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Unknotting number of different knots 16 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Unknotting number of different knots If we change {8, 11, 12, 16, 18, 20} crossings in K(5, 6), we get n1414274. So u(n1414274) = 4 16 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Unknotting number of different knots If we change {8, 11, 12, 16, 18, 20} crossings in K(5, 6), we get n1414274. So u(n1414274) = 4 Knot Programs LinKnot (S. Jablan, R. Sazdanovic) and Knotscape (Jim Hoste and Morwen Thistlethwaite) to identify the knots in knot table (Hoste-Thistlethwaite Knot Table). 16 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 17 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results If we change {8, 11, 12, 15, 16, 20} crossings in K(5, 6), we get n1418351. So u(n1418351) = 4 17 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results If we change {8, 11, 12, 16, 20} crossings in K(5, 6), we get n1424498. So u(n1424498) = 5 18 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results If we change {8, 11, 12, 14, 16, 19, 20} crossings in K(5, 6), we get 93. So u(93) = 3 19 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Unknotting Torus Knots Minimal Unknotting Crossing Data for Torus Knots Unknotting number of some knots Bibliography Results 20 / 23 Objective Minimal Unknotting Crossing Data for Torus Knots Bibliography A. Kawauchi, A survey of knot theory. Translated and revised from the 1990 Japanese original by the author. Birkhuser¨ Verlag, Basel, 1996. Ayaka Shimizu, Region crossing change is an unknotting operation. to be appear in J. Math. Soc. Japn, math.GT/1011.6304v3, 2012. Cheng Zhiyun, When is region crossing change an unknotting operation?, Math. Proc. Cambridge Philos. Soc., 155, 257-269, 2012. V. Siwach and P. Madeti, Unknotting torus knots: A new approach, Communicated. P. Kronheimer and T. Mrowka, Gauge theory for embedded surfaces, I, Topology, 32, 773-826. 1993. P. Kronheimer and T. Mrowka, Gauge theory for embedded surfaces, II, Topology, 34, 37-97, 1995. V. Siwach and P. Madeti, A sharp upper bound for region unknotting number of torus knots, J. Knot Theory Ramifications, 22, 1-21, 2013.
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