Knots, Crossing Changes, and Surfaces in 4-Dimendions

Home , Slice knot, Unknot, Unknotting number

Knots, Crossing Changes, and Surfaces in 4-dimendions Kayla Davie, Brendan Owens PhD University of Glasgow, Department of Mathematics, Glasgow, Scotland Spelman College, G-STEM, Math RaMP

ABSTRACT RESULTS CONCLUSION Knot theory is a branch of topology that focuses on the The question we wished to answer is: What are the slicing numbers (minimum number of crossing changes needed to convert to a slice knot) With this research, we: study of mathematical knots. We study 11 crossing of each 11 crossing knot? knots and knot invariants related to smoothly immersed Found slicing numbers for 416 of 552 11 crossing knots. disks in the four-ball. The knot invariant we focus on is A crossing change is defined in terms of a knot diagram by changing the over- and understrand of a single crossing in the diagram. slicing number, which is the minimal number of Found 4-ball crossing numbers for 416 11 crossing knots. crossing changes to a slice knot. Using known A knot projection is a projection of the knot into the plane relationships between knot invariants, computer which every place where the knot crosses itself also indicates Found the smooth 4-genus previously unknown for 71 11 applications used to manipulate knots, and other which strand would be above the other if the knot were in crossing knots. advanced programming, we were able to determine the 3-dimensional space. slicing number for 414 11 crossing knots (out of a total By completing this research, we found important knot of 552). We also obtained information on the 4-ball A knot invariant is a quantity defined for a knot that is the invariants for most 11 crossing knots. While we did not find crossing number (minimal number of crossing changes FIGURE 2. Two knot projections of Stevedore’s knot (6 ). same for equivalent knots. 1 the slicing number and 4-ball crossing number for all 11 in a movie of a disk bounded by the knot) and the crossing knots, the ones we did not find are perhaps the smooth 4-genus (minimum genus of a smooth surface FINDING SLICING NUMBERS most important because they shed light on knots that will embedded in the 4-ball with boundary the knot) of need new techniques to be analyzed by mathematicians in many of these knots. The knot invariants we analyze include: the future.

INTRODUCTION unknotting number, u: minimum number of crossing changes to unknot slicing number, u : minimum number of crossing changes to slice knot Knot theory is a branch of mathematics that focuses on s FUTURE RESEARCH 4-ball crossing number, c*: minimum number of crossing changes in a movie the study of knots and their properties. Mathematical In order to find slicing numbers, we began by checking: of a disk bounded by the knot For this research, we focused on 11 crossing knots and the study of knots began in the 19th century and since then smooth 4-genus, g*: minimum genus of an embedded surface in [a, b] knot invariants associated with these knots, however, there mathematicians have discovered and studied various bounded by the knot are several higher crossing knots with even less knot properties that are distinct to given knots. These invariants known about them. We can further this research properties are called knot invariants and include the by beginning to analyze 12 crossing knots. unknotting number, slicing number, 4-ball crossing For knots that did not have invariants satisfying number, smooth 4-genus, signature and others. any of the above conditions, we used a program called We also did not find the results we were seeking for all 11

Kirby Calculator to manually test the crossing changes crossing knots so future research could include finding J. C. Cha and C. Livingston’s Table of knot invariants needed to convert a knot to slice. new ways to analyze the 11 crossing knots we were not serves as a database for currently known information able to find invariants for. about many knot invariants. The table has information The figure to the right shows that the knot 11a13 can be for all knots with up to 12 crossings, however, some converted into the knot 6 , a slice knot, using one knot invariants for knots with 11 and 12 crossings are 1 ACKNOWLEDGMENTS crossing change. Therefore, the slicing number for this not as well-known as for those with 10 or fewer knot is 1. crossings. In their paper, Immersed disks, slicing We would like to thank the University of Glasgow

numbers, and concordance unknotting numbers, B. Mathematics Department, Spelman College Mathematics

Owens and S. Strle focused on three knot invariants Department, and Arcadia University Department of Global For knots that could not be manually tested very related to smoothly immersed disks in the four-ball: 4- Studies. Additionally, we would like to thank the National easily, we tested them using Maple and Matematica ball crossing number, slicing number, and concordance Science Foundation for funding this research through the programs based on a theorem created by B. Owens and unknotting number. Their research was used to provide Spelman Global STEM program (G-STEM, Award ID:HRD S. Strle with the help of Dr. Brendan Owens. FIGURE 3. Converting 11a13 into a slice knot with one crossing change using Kirby Calculator. information listed in Table of knot invariants for knots 0963629); the U.S. Department of Education; Student Aid with at most 10 crossings. and Fiscal Responsibility Act; Title III Grant (SAFRA, Part F). Any opinions, findings and conclusions or The goal of this project was to compute some of these KNOT INVARIANT TABLES recommendations expressed in this material are those of the invariants for as many as possible of the 552 knots with authors and do not necessarily reflect the views of the NSF or the U.S. Department of Education. 11 crossings. We focus mainly on the slicing number, Below are two of five tables of knot invariants that were compiled during this research. These show the findings for 184 knots. which is the minimal number of crossing changes required to convert a knot into a slice knot. REFERENCES

1. A.G. Tristram. Some cobordism invariants for links. Proc. Cambridge Philos. Soc., 66:251-264, 1969. BACKGROUND 2. B. Owens and S. Strle. Immersed disks, slicing numbers and concordance unknotting numbers. arXiv:1311.6702, 2013. A knot is a smooth isotopy class of embedded closed 3. C.C. Adams, The knot book, American Mathematical non-self-intersecting curves in R3 (or S3). Society, Providence, RI, 2004, An elementary A knot K in S3 is called slice if it bounds a smoothly introduction to the mathematical theory of knots, Revised properly embedded D2 in D4. reprint of the 1994 original.

4. C. Livingston. Knot theory, volume 24 of Carus Mathematical Monographs. Mathematical Association of America, Washington, DC, 1993. 5. F. Swenton. Kirby calculator. http://www.kirbycalculator.net 6. I. Stewart. Concepts of modern mathematics. Courier Dover Publications, 1995. 7. K. Murasugi. On a certain numerical invariant of link types. Trans. Amer. Math. Soc., 117:387-422, 1965. 8. J.C. Cha and C. Livingston. Table of knot invariants. http://www.indiana.edu/~knotinfo. 9. J. R. Weeks. The shape of space. CRC press, 2001. FIGURE 1. Movie of a slice disk for Stevedore’s knot (61) which is a slice knot.