(Construct Surfaces from Knots) Hun Kim, Kyoung Il Park and R.Jooyoung Park Dept. of Mathematics and Computer Science, Korea Science Academy of KAIST 105 47 Baekyanggwanmun-ro, Busanjin-gu, Busan 614 100, Korea HunKim@kaist.ac.kr mathloveman@hanmail.net Abstract 3 S 1 R 3 2 A Knot, in the language of mathematics, is an embedding of a circle S 1 into Euclidean 3-space, R 3. And a knot, as a simple closed curve in R 3, is the boundary of an orientable surface called a Seifert surface. In this workshop, participants will make Seifert surface from a knot using color cray. In fact, any orientable compact surface can be classified by calculating Euler characteristic and counting number of holes. So, participants will calculate Euler characteristic using a graph on the Seifert surface to classifing surface. (Knots) 3 S 1 R 3 1: [9] R 3
2: [9] 3 3: [9] 4 3 1, 3 (Seifert Surfaces) 3
(a) 1 (b) 2 (c) 3 4: [9] 5: 6 = = 6: 180 180
= = 7: = = 8: (Classification of Surfaces) 9: v e + f v e f 10 11,
10: v = 11, e = 14, f = 5 14, 5 v e + f = 11 14 + 5 = 2 2 2 2 2 2 2 2 11 = = = 11: = 12: 13 0
13: v e + f = 5 10 + 5 = 0 2 14 n n 0 1 14 2 (a) (b) (c) 14: 15 1 2 # 15: 3 3 v e + f = (10 3) (20 3) + (10 2) = 2 n v e + f = (n 5 (n 1) 3) (n 10 (n 1) 3) + (n 5 2 (n 1)) = 2 2n (1) k n k (1)
k k n v e + f = (n 5 (n 1) 3) (n 10 (n 1) 3) + (n 5 2 (n 1)) k = 2 2n k (2) 16 16: 16 v e + f = 8 13 + 3 = 2 2 n (2) 2 2n k = 2 2n 2 = 2 1 16 (Activity)
= = 그림 17: 활동 참고문헌 [1] Colin C. Adams. The Knot Book: An Elementary Introdcution to the Mathematical Theory of Knots. AMS, 2001. [2] Gerhard Burde and Heiner Zieschang. Knots. Walter de Gruyter, 2010. [3] J.C. Cha and C. Livingston. Knotinfo: Table of knot invariants. http://www.indiana.edu/ knotinfo. [4] David W. Farmer and Theodore B. Stanford. Knots and Surfaces: A Guide to Discovering Mathematics, volume 6 of Mathematical World. AMS, 1996. [5] Louis H. Kauffman. On Knots. Princeton University Press, 1987. [6] Dale Rolfsen. Knots and Links. AMS CHELSEA PUBLISHING, 2003. [7] Mereke van Garderen and Jarke J. van Wijk. Seifert surfaces with minimal genus. Proceedings of Bridges 2013, 2013. [8] Jarke J. van Wijk and Arjeh M. Cohen. Visualization of seifert surfaces. IEEE Trans. on Visualization and Computer Graphics, 12(4), 2006. [9] Wikipedia. Knot theory. http://en.wikipedia.org/wiki/knot_theory.