Folded Ribbon Knots

and Ribbonlength

John Carr Haden and Troy Larsen

SUMS 2020

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What is a Knot?

• Tie a knot (like you would tie your shoelaces, for example)
• Join the ends so that it can’t be undone!
• A mathematical knot, K, is a simple, closed curve in three-dimensional space

https://www.pinterest.com/pin/568227677978603378/

https://mathcurve.com/courbes3d.gb/noeuds/huit_ferreol.gif

What is a Knot? (cont.)

• Knot Diagrams: projections with over under information preserved
• If we manipulate a knot in space, its diagram changes!
• Two knots are equivalent if and only if one of their diagrams can be transformed into the other through a finite series of manipulations

K. Murasugi, Knot Theory and its Applications, Birkhäuser, Boston, 1996.

Trefoil

Crossing Number: Cr(K)

• Tells us the minimal number of crossings over all possible diagrams of K
• Not the same as the number of crossings in one diagram!
• Helps to distinguish between knots

Unknot

Torus Knots

• Can be embedded onto a torus (no self-intersections)
• Denoted T_p,q
• p is the number of times around the longitude
• q is the number of times around the meridian
• Proven crossing number: Cr(K) = min{ p(q-1), q(p-1) }

https://www.researchgate.net/figure/A-trefoil-knot-embedded-on-a-torus_fig2_314797927

Trefoil T_2,3

Making a Torus Knot Diagram

• p is the number of strands (longitudes)
• q is the number of wrappings (meridians)
• Creating a diagram of a Torus knot in the plane:
• Lay p > 0 strands horizontally parallel
• Wrap the top (if q < 0) or bottom (if q >0) strand over all others
• Repeat until q strands have been wrapped
• Connect the top left end to the top right end, etc.

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T_5,-3

What is a Folded Ribbon Knot?

• A folded ribbon knot, K, is created from a rectangular ribbon of fixed width w.
• Ribbon constructed around a knot diagram (with folds).

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B. Kennedy, T.W. Mattman, R. Raya, D Tating, 2008; Ribbonlength of torus knots. J. Knot Theory Ramifications. 17 no. 1, 13-23.

E. Denne, Folded Ribbon Knots in the Plane, 2019.

T_2,5

Trefoil (Almost)

Ribbonlength: Rib(K)

• Rib(K) = Length(K) / width
• Rib(K) = 6.88 for the trefoil knot (to two digits)
• Goal 1: minimize ribbonlength for any knot
• Goal 2: relate minimal ribbonlength to crossing number

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c₁Cr(K)^⍺ ≤ Rib(K) ≤ c₂Cr(K)^𝛽

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• Conjecture: ⍺ = ½, 𝛽 = 1
• Prof. Denne proved 𝛽 = 1.5 (2019)
• We constructed a family of torus knots where ⍺ = ½ (first ever!)

Folded Ribbon Torus Knots

• We assume that p (number of strands) > q (number of wrappings)
• Here we have the trefoil, T_3,2

How We Compute Ribbonlength

• Our construction is comprised of squares and triangles
• We let width be 1
• Unit square has ribbonlength 1; unit right triangle has ribbonlength ½

Folding q Strands Up and Over

• For T_3,2 we have 2x2 triangles and 2 squares
• In general, 2q triangles in front, q squares in back

Making Horizontal Connections

• For T_3,2 we have 2(1)=2(3-2) triangles
• In general, 2(p-q) triangles

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Finishing the Construction

• For T_3,2 we create 2 triangles in front and remove 2 triangles from the square
• In general, q triangles added in front, q triangles removed from square

Computing the Ribbonlength

• Our constructions creates:
• q strands that we fold up and over: 2 triangles, 1 square each
• p - q horizontal connections: 2 triangles each
• q vertical connections: +1 triangle, -1 triangle each
• Adding them together: 2q(½) + q(1) + 2(p-q)(½) + q(½-½) = 2q + p - q
• Ribbonlength: Rib(T_p,q) ≤ p + q

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• For the trefoil, T_3,2 , we see Rib(T_3,2) = 5 < 6.88

Showing ⍺ = ½

• Goal: find c₁ such that p + q = Rib(T_p,q) ≤ c₁ Cr(T_p,q)^½
• We first square both sides and isolate c₁:

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(p + q)² / Cr(T_p,q) ≤ (c₁)²

*insert horrible algebra here*

c₁= 2√2 (p/q).

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• So our general formula reads: Rib(T_p,q) ≤ 2√2 (p/q) Cr(T_p,q)^½

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Showing ⍺ = ½ (cont.)

• Our general formula reads: Rib(T_p,q) ≤ 2√2 (p/q) Cr(T_p,q)^½
• For families where p = aq + b, we have

Rib(T_p,q) ≤ 2√2 (a + ½ b) Cr(T_p,q)^½

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• An example: T_q+1,q yields a = 1, b = 1. So,

Rib(T_q+1,q) ≤ 2√2 (1 + ½) Cr(T_q+1,q)^½

Rib(T_q+1,q) ≤ 2√2 (3/2) Cr(T_q+1,q)^½

Rib(T_q+1,q) ≤ 3√2 Cr(T_q+1,q)^½

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Other Things We Did

• Recall conjecture: c₁Cr(K)^𝛼 ≤ Rib(K) ≤ c₂Cr(K)^𝛽
• 𝛼 = ½ and 𝛽 = 1
• Other constructions that we made:
• T_2,p torus knots (different construction)
• Pretzel knots (first ever!)
• Twist knots (we win for small crossing number)
• Rational knots (first ever!)
• All have 𝛽 = 1 giving evidence for the conjecture holding for all K.

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Our T_2,p Result: Rib(T_2,p) ≤ 2p = 2Cr(T_2,p)

T_2,5

T_2,3

Our Twist Knot Result: Rib(T_n) ≤ 2Cr(T_n) + 2

T₃

T₅

Our Pretzel Knot Result: Rib(P_p,q,r) ≤ 2Cr(P_p,q,r) + 2

P_3,3,1

Our Rational Knot Result: Rib(C_n) ≤ 6Cr(C_n) - 2

https://doi.org/10.1063/1.4900924

C[2,3,2]

Acknowledgements

• Washington and Lee University for SRS Funding
• Professor Denne for mentorship and supervision
• Allison Young ‘20 and Corinne Joireman ‘21 for their insights

Questions?