Straight Number and Volume

Transcription

1 October 13, 2018 nick.owad.org

2 Knots and Diagrams Basics A knot is an embedded circle in S 3. A knot diagram is a projection into 2 dimensions.

3 Knots and Diagrams Straight Diagram A knot diagram is straight if it looks like this:

4 Knots and Diagrams Straight Diagrams One should ask the following question: Question (S. Jablan and L. Radović, 2015) What knots have straight diagrams? Theorem (O. 2018) Every knot has a straight diagram.

5 Knots and Diagrams Algorithm This image partially comes from Adams, Shinjo, and Tanaka.

6 Knots and Diagrams Straight Number Since every knot has a straight diagram, we can make an invariant. Definition The straight number of a knot K, str(k), is the minimum number of crossings over all straight diagrams of K.

7 Knots and Diagrams Straight Code The knot can now be defined by the string of integers ±1, ±2,..., ±n, positive for crossing over the straight strand, and negative for under ( 7, 6, 1, 4, 3, 2, 5, 8, 9)

8 Diagrammatic Straight Number Quick results There are a finite number of knots with a fixed straight number, str(k) = n. The quick upper bound on this number is n!2 n. Also, by definition, c(k) str(k). Definition A knot K is perfectly straight if str(k) = c(k).

9 Diagrammatic Straight Number Quick results Theorem (O., 2018) 1 Every torus knot T 2,q is perfectly straight. 2 Every n-pretzel knot is perfectly straight. 3 Every 2-bridge knot K p/q where the continued fraction of p/q has length less than 6 is perfectly straight. 4 Every knot with 7 or less crossings is perfectly straight.

10 Diagrammatic n-pretzel knots are perfectly straight A standard diagram of the ( 3, 5, 3, 2)-pretzel knot:

11 Diagrammatic Straight Diagram A straight diagram of the ( 3, 5, 3, 2)-pretzel knot:

12 Diagrammatic Families of knots which are not perfectly straight But now you should ask, What are some knots which are not perfectly straight? The first knot which is not perfectly straight in the standard Rolfsen table is 8 16!

13 Diagrammatic Knots which are not perfectly straight How do you prove that? Brute Force! We check every possible straight diagram that has eight crossings. Turns out that 8 18 also is not perfectly straight.

14 Diagrammatic 8 16 and 8 18 Figure: Images courtesy of Knotinfo - knotinfo/

15 Diagrammatic str(8 16 ) = 10

16 Diagrammatic str(8 18 ) = 10

17 Diagrammatic The straight number of all knots with 10 or less crossings Theorem (O. 2018) Let K be a knot with 10 or less crossings. Then K is perfectly straight or str(k) is given below. str(8 α) = str(9 β ) = 10, α {16, 18}, β {32, 47} str(9 γ) = str(10 δ ) = 11, γ {29, 33, 34, 41}, δ {69, 75, 97, 101, 165} str(9 40) = str(10 ɛ) = 12, ɛ [84, 89] [91, 93] {96, 100} [103, 105] {108} [111, 114] [116, 119] {122, 123} str(10 ζ ) = 13, ζ {102, 121}

18 Volume s relation to straight number How do you prove that and why volume? We used SnapPy to check all the thousands of possibilities. Sometimes SnapPy fails at identifying a knot. Usually this is the unknot or the trefoil, but we need to be sure if we are checking every knot. SnapPy is good at finding the volume with a high accuracy, so lets use that!

19 Volume s relation to straight number Algorithm Make a list of all straight codes with n crossings and check that they are realizable. Use SnapPy to try to identify them all. If they are identified, we are done with them, put them in a list D. If they are not identified, put them in another list, R. Let K n be all knots with n crossings or less. Then D K n. And we note the smallest volume, v, of the knots in K n \ D. Then we compute the volume of every knot from R. If the volume of every knot in R is less than v, we are done!

20 Volume s relation to straight number What does this have to do with Volume? This technique depends on the knots we are missing having a higher volume than the knots which often are not identified. For each n we tested, the knots we did not find were always the higher volume knots!

21 Volume s relation to straight number Straight Number vs Volume for c(k) =

22 Volume s relation to straight number Straight Number vs Volume for c(k) =

23 Volume s relation to straight number Straight Number vs Volume for c(k) =

24 Volume s relation to straight number Straight Number vs Volume for c(k) =

25 Volume s relation to straight number Straight Number vs Volume for c(k) =

26 Volume s relation to straight number Highest Volume Straight knots What can we say about this apparent relation? Observationally, it appears that a knot K with str(k) > c(k) will have higher volume also. But this is very hard to make precise. For instance, there are high volume knots which are perfectly straight. And some lower volume knots which are not perfectly straight. This means we should probably look at a random straight knot. Which we won t have time to talk about today.

27 Volume s relation to straight number Highest Volume Straight knots Another option is to just look at the highest volume knots with str(k) = n. Some of these seem to have a common pattern. For example, the knot is the fourth highest volume knot with 11 crossings, and the highest volume straight number 11 knot. So is a perfectly straight knot. Thus, it does not fall into our ideas from the last slide.

28 Volume s relation to straight number

29 Volume s relation to straight number Snail? Figure: Image courtesy of wikipedia.org

30 Volume s relation to straight number Snail Links Blame Josh Howie for Snail We create a two parameter family of links, Snail(s, c): s semicircles c crossings In this case, we are looking at Snail(3, 13).

31 Volume s relation to straight number Snail Links Then we connect it up in the following way:

32 Things we need to investigate further Quick snail results Given c, the number of crossings, then Snail(s, c) is defined for 2 s c+1 2. The bridge number: b(snail(s, c)) s. Moreover, when s = 2, and a { c+2 3, c+3 3 } is an integer, then Snail(2, c) = Snail(a, c) and it has bridge number 2. With partial fraction decomposition [2, 1, 1,..., 1, 1, 2]. Also, when c is odd, Snail( c+1 2, c) is also 2-bridge, with partial fraction decomposition[3, 2, 2,..., 2, 2].

33 Things we need to investigate further Look at the volume of the first 249,500 Snail links s c 1000 Vol = 0

34 s = c+1 2 s = c+2 3, c+3 3 s c 4 s c 5 s = 2

35

36 Things we need to investigate further 2-bridge snail links In progress work with Jessica Purcell We know that there are three of these are 2-bridge knots. Are there any others? Conjecture Given c, if s 2, c+2 3, c+3 3 c+1, or 2, then b(snail(s, c)) 3. We have checked this with SnapPy and these are the only 2-bridge knots it has found for c < 100.

37 Things we need to investigate further When are two snails the same? In progress work with Jessica Purcell What knots are the same for a given c? Question Given c, for what pairs (s, s ), does Snail(s, c) = Snail(s, c)? For example, when c = 50, SnapPy tells us that the following s values are the same links. (3, 21), (4, 15), (6, 19), (7, 24), (10, 22), (12, 16), (13, 25). For c = 51: (2, 18), (3, 11), (4, 8), (5, 12), (6, 10), (9, 25), (15, 22), (16, 24), (17, 21), (19, 23)

38 Things we need to investigate further Highest volume knots by crossing In progress work with Jessica Purcell Champanerkar, Kofman, and Purcell note in their paper Volume bounds for weaving knots, that Xiao-Song Lin conjectured that Weaving links are the highest volume links for fixed crossing number. But in work currently being done with Jessica Purcell, we have found numerical evidence that Snail links have higher volumes.

39 Things we need to investigate further Highest volume knots by crossing In progress work with Jessica Purcell Let LVW (c) be the largest volume of a weaving link with c crossings, and LVS(c) be the largest volume of a snail link with c crossings. For c [2, 56], LVW (c) > LVS(c) 22 times, and the other 34 times, LVS(c) > LVW (c). For c [57, 200], LVS(c) > LVW (c).

40 Things we need to investigate further Questions Some questions from the topics discussed: 1 What is the straight number of Weaving knots? 2 Can we improve the upper bound on straight number? 3 Can we find a useful lower bound? 4 Is it possible to characterize what makes a knot perfectly straight from geometric or topological perspectives?

41 Things we need to investigate further Connected sum Conjecture The straight number is additive under connect sum, i.e. str(k#j) = str(k) + str(j).

42 Thanks Thank you

43 Thanks References C. Adams, R. Shinjo and K. Tanaka, Complementary Regions for Knot and Link Complements, Annals of Combinatorics. 15, (October 2011), no. 4, A. Champanerkar, I. Kofman., and J. Purcell, Volume bounds for weaving knots, Algebraic and Geometric Topology, 16 (2016), No. 6, M. Culler, N. M. Dunfield, M. Goerner, and J. R. Weeks, SnapPy, a computer program for studying the geometry and topology of 3-manifolds, S. Jablan and L. Radović, Meander Knots and Links, Filomat (29)(10) (2015) pp M. Lackenby, The volume of hyperbolic alternating link complements. With an appendix by I. Agol and D. Thurston, Proc. London Math. Soc. 88 (2004),