The Hexagonal Lattice Number of the Figure Eight is 11 2019 Nebraska Conference for Undergraduate Women in Mathematics Mia Nguyen University of Nebraska-Lincoln, Department of Mathematics Quick review of knot theory are able to make some estimates about the stick number of some simple knots. From the cubic model of the knot, we project it Knot theory is a branch of topology that studies three- onto the xyw-plane and transform it to hexagonal lattice. For dimensional manifolds. A mathematical knot is a closed curve the conversion, the angle of 30o at each corner and the minimal that is embedded in 3-dimensional Euclidean space. Two or number of sticks are prominent. more knots combined together are considered as a link. It is not obvious to determine if 2 given knots are equivalent to each other or not. Proving stick number of the figure eight is Definition of hexagonal lattice 11 Hexagonal lattice includes points such that an equilateral trian- gle is formed by every 3 nearby points. There are 4 orientations associated with the lattice, one goes up, one goes to the right, and 2 oblique axes. The simple hexagonal lattice is defined as the point lattice where x = <1, 0, 0>, y = <1/2, 3/2, 0>, w = <0, 0, 1>, and z = y - x. Figure 5: The 52 knot and its projection in the simple hexagonal lattice. The x-stick, y-, z-, and w-sticks are straight line segments that are parallel to directions of x, y, z, and w. Stick number is the In Figure 5, jP jx = 2; jP jy = 4; jP jz = 3; jP jw = 5, and smallest number of edges needed to form a knot. There are not jP j = 14. many knots that their stick numbers are achieved exactly be- cause it is hard to determine if we reach the minimal number of stick. jP jx; jP jy; jP jz, and jP jw will denote the number of x- sticks, y-sticks, z-sticks, and w-sticks, respectively, and jP j will Figure 2: The 41 knot and its minimal stick conformation in the simple cubic show the total number of sticks. lattice. Figure 6: The 61 knot and its projection in the simple hexagonal lattice. In Figure 6, jP jx = 3; jP jy = 2; jP jz = 4; jP jw = 7, and jP j = 16. Figure 3: The projection of the figure eight in the simple hexagonal lattice. Figure 1: The directions of axes in the simple hexagonal lattice. In Figure 3, jP jx = 2; jP jy = 2; jP jz = 3; jP jw = 4; and jP j = 11. Theorem about stick number Theorem 1 Conclusions For any knot K in the simple hexagonal lattice, s[K] ≥ 5b[K]. We know that lower bound for stick numbers of any knot in Knots with the bridge index equal to 2 have different stick num- the simple hexagonal lattice is equal or greater than 5b[K] where bers. The figure eight and trefoil are the only 2 knots that we b[K] is the bridge number of the knot K (Theorem 1). Since the could prove the minimal number of sticks is 11. Figure 7: The 62 knot and its projection in the simple hexagonal lattice. bridge number of trefoil is 2 with 2 local minima and 2 local maxima, its stick number should be at least 10. In Figure 7, jP jx = 3; jP jy = 3; jP jz = 2; jP jw = 7, and jP j = 15. Theorem 2 Further work In the simple hexagonal lattice, the stick number of any non- References trivial knot is at least 11. [1] Ryan Bailey, Hans Chaumont, Melanie Dennis, Jennifer We need at least 10 sticks to construct any nontrivial knot (The- McLoud-Mann, Elise McMahon, Sara Melvin, and Geoffrey orem 1). However, it was shown that any knot built with 10 Schuette. Stick numbers in the simple hexagonal lattice. In- sticks is trivial. That leads to a conjectured new lower bound for volve, a Journal of Mathematics, pages 503–512, 2015. stick number. [2] C. E. Mann, J. C. McLoud-Mann, and D. P. Milan. The stick number for the simple hexagonal lattice. J. Knot Theory Conjecture Ramifications, pages 14–21, 2012. For any knot K in the simple hexagonal lattice, Acknowledgements I gratefully acknowledge the very helpful discussions pro- Figure 4: The 51 knot and its projection in the simple hexagonal lattice. s[K] ≥ 5b[K] + 1 vided by Professor Alex Zupan, my research supervisor, for In my research project, I proved that the hexagonal lattice num- In Figure 4, jP jx = 2; jP jy = 3; jP jz = 2; jP jw = 6, and his enthusiasm, valuable support, and encouragement of this ber of the figure eight is also 11. Based on bridge number, we jP j = 13. research work..