Diagonal Diagrams: An alternative representation for torus links

Transcription

1 : An alternative representation for torus links Westminster College September 11, 2012

2 What is a knot?

3 What is a knot? A knot is a closed curve in three dimensional space. Knots may be represented as a projection in two dimensional space, called a knot diagram. A knot of more than one component is called a link.

4 What is a knot?

5 Overview What is a knot? Rotation Proposition Unknot Proposition Linked Unknots Proposition Greatest Common Divisor Theorem Torus Link Theorem Future Work and s

6 Grid diagrams must be square. There must be an X in every row and column. There must be an O in every row and column. Vertical connections are over every horizontal connection.

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9 The three Cromwell moves preserve a grid diagram: 1. C 1, torus translation 2. C 2, commutation 3. C 3, stabilization

10 An example of C 1, torus translation:

11 An example of C 2, commutation:

12 An example of C 3, stabilization:

13 The X s of a diagonal diagram are on the main diagonal. The Os are divided between two rows parallel to the diagonal. This diagonal diagram is D 5,2.

14 For the diagonal diagram given by D x,y Dx,y is on an x by x grid. Dx,y has y Os to the left of the main diagonal. D 7,3 D 6,2

15 D 8,1

16 D 8,1 D 6,3

17 D 8,1 D 6,3 D 7,2

18 D 8,1 D 6,3 D 7,2 D 8,4

19 Rotation Proposition Proposition: D x,y = D x,x y.

20 Rotation Proposition Proposition: D x,y = D x,x y. D 5,2 = D 5,3

21 Unknot Proposition Proposition: Every diagonal diagram of the form D x,1 represents the unknot. D 6,1 D 7,1 D 8,1

22 Unknot Proposition There are no crossings between an adjacent X and O. There are no crossings on the edge of a grid diagram. D 5,1

23 Linked Unknots Proposition Proposition: Let x be even. Then D x, x 2 is an x 2 -component link in which every component is an unknot linked with every other component of the link. D 4,2 is the hopf link:

24 Linked Unknots Proposition D 8,4 is also a set of linked unknots:

25 The GCD The greatest common divisor is the largest number that divides two numbers evenly. gcd(1, 6) = 1 gcd(6, 15) = 3 gcd(7, 4) = 1 gcd(2, 10) = 2 gcd(8, 12) = 4 gcd(24, 9) = 3

26 GCD Theorem Theorem: The diagonal diagram D x,y is composed of gcd(x, y) components, each of the form D x y gcd(x,y),. gcd(x,y)

27 GCD Theorem Theorem: The diagonal diagram D x,y is composed of gcd(x, y) components, each of the form D x y Example: D 10,4 gcd(10, 4) = 2 gcd(x,y),. gcd(x,y)

28 GCD Theorem Theorem: The diagonal diagram D x,y is composed of gcd(x, y) components, each of the form D x y Example: D 10,4 gcd(10, 4) = 2 D 10,4 has two components gcd(x,y),. gcd(x,y) D 10,4

29 GCD Theorem Theorem: The diagonal diagram D x,y is composed of gcd(x, y) components, each of the form D x y Example: D 10,4 gcd(10, 4) = 2 D 10,4 has two components gcd(x,y),. gcd(x,y) Each component is a copy of D 5,2 D 10,4

30 GCD Theorem Theorem: The diagonal diagram D x,y is composed of gcd(x, y) components, each of the form D x y Example: D 10,4 gcd(10, 4) = 2 D 10,4 has two components gcd(x,y),. gcd(x,y) Each component is a copy of D 5,2 D 10,4

31 GCD Theorem Theorem: The diagonal diagram D x,y is composed of gcd(x, y) components, each of the form D x y Example: D 10,4 gcd(10, 4) = 2 D 10,4 has two components gcd(x,y),. gcd(x,y) Each component is a copy of D 5,2 D 10,4 D 5,2

32 GCD Theorem Example: D 24,9

33 GCD Theorem Example: D 24,9 gcd(24, 9) = 3

34 GCD Theorem Example: D 24,9 gcd(24, 9) = 3 D 24,9 has three components D 10,4

35 GCD Theorem Example: D 24,9 gcd(24, 9) = 3 D 24,9 has three components Each component is a copy of D 8,3 D 10,4

36 GCD Theorem Example: D 24,9 gcd(24, 9) = 3 D 24,9 has three components Each component is a copy of D 8,3 D 10,4 D 8,3

37 GCD Theorem Reasoning: Consider D 12,8, which is 4 copies of D 3,1. Begin with X at (1, 1)

38 GCD Theorem Reasoning: Consider D 12,8, which is 4 copies of D 3,1. The X at (1, 1) is horizontally connected to an O at (1, 9 (1 + 8) mod 12) The O at (1, 9) is vertically connected to the X at (9, 9)

39 GCD Theorem Reasoning: Consider D 12,8, which is 4 copies of D 3,1. The X at (9, 9) is horizontally connected to an O at (9, 5 (9 + 8) mod 12) The O at (9, 5) is vertically connected to the X at (5, 5)

40 GCD Theorem Reasoning: Consider D 12,8, which is 4 copies of D 3,1. The X at (5, 5) is horizontally connected to an O at (5, 1 (5 + 8) mod 12) The O at (5, 1) is vertically connected to the X at (1, 1)

41 GCD Theorem Reasoning: Consider D 12,8, which is 4 copies of D 3,1. We have returned to the X at (1, 1) The columns 1, 5, and 9 are connected

42 Torus Links A link that has no crossings when lying on the surface of an unknotted torus is called a torus link. The torus link T (3, 4):

43 Torus Links We can think of the arbitrary torus link T (p, q) as p twists on q strings. An arbitrary torus link:

44 Torus Links Furthermore, it is worth noting that any torus link T (p, q) has several nice properties: 1. T (p, q) = T (q, p). 2. T (p, q) has one twist box. 3. T (p, q) has gcd(p, q + p) components. 4. If gcd(p, q) = 1, T (p, q) is a prime knot. 5. The crossing number of T (p, q) = q(p 1). 6. T (p, q) is (p 1)-alternating and (q 1)-alternating.

45 Torus Link Theorem Theorem: The diagonal diagram D x,y is the torus link T (y, x y). Proof :

46 Torus Link Theorem D 7,3 is the torus link T (3, 4):

47 s and Future Work Other results on diagonal diagrams k-alternating links Cromwell moves on a grid Other families of grid diagrams

48 Acknowledgements Dr. Jeffrey Boerner Dr. David Offner Student Government Office Student Club Room Coffee

49 References Adams, C. C. (1994). The knot book: an elementary introduction to the mathematical theory of knots. New York: W.H. Freeman. Atiyah, M. F. (1990). The geometry and physics of knots. New York: Cambridge University Press. Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, Cha, J. C. & Livingston, C. (2011). KnotInfo:Table of Knot Invariants. knotinfo Cromwell, P. R. (2004). Knots and links. New York: Cambridge University Press. Cromwell, P. R. (1995). Embedding knots and links in an open book I: Basic properties. Topology and its Applications. 64, Hackney, P., Van Wyk, L., & Walters, N. (2004). k-alternating knots. Retrieved from vanwyk/res/kaltknots taia.pdf Jin, G. T. & Lee, H. J. (2011). Prime knots whose arc index is smaller than the crossing number. arxiv.org. Jin, G. T. & Park, W. K. (2010). A tabulation of prime knots up to arc index 11. arxiv.org. Jin, G. T. & Park, W. K. (2010). Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots. arxive.org. Lickorish, W. B. R. (1997). An introduction to knot theory. New York: Springer. Liedy, C. (2008). Project 2: grid diagrams and grid moves. Retrieved from wesleyan.edu/home/cleidy/web/teaching/242fall2008/project2.pdf Livingston, C. (1993). Knot theory. Washington, D.C.: Mathematical Association of America. Murasugi, K. (1996). Knot theory and its applications. Boston: Birkhäuser. Rolfsen, D. (1976). Knots and links. Berkley, CA: Publish or Perish.