Lower Bounds for the Ropelength of Reduced Knot Diagrams

by:Robert McGuigan

Research Experience for Undergrads

• What: A (typically summer) program where undergraduate students experience what it is like to conduct research in mathematics (or other fields).

• Who: Any student who will still be an undergraduate the following fall.

• Where: All over the US (and abroad). Many are funded by the National Science Foundation (NSF).

Research Experience for Undergrads

• Why:– Great for deciding if graduate school is right for

you (or simply preparing for it).– An excellent addition to your resumé.– Nothing beats getting paid to do math!

Research Experience for Undergrads

• At my REU at CSU San Bernardino:– I worked closely with a professor studying

ropelength, a topic in knot thoery.– I authored my first mathematical paper.– I created an accompanying poster.

Terminology

• A conformation is a smooth, closed curve in 3-space which does not intersect itself.

• Two conformations are said to be of the same knot type if one can be “deformed” into the other without passing the curve through itself.

Terminology

• A conformation is typically presented as a knot diagram, which projects the conformation onto the xy-plane.

• The knot diagram can be divided into maximal overpasses and underpasses.

Division into Maximal Arcs

Labelling the Maximal Overpasses and Underpasses

• We will let pi denote the point with maximum height on the ith maximal overpass encountered as one traverses the knot from some starting point.

• Similarly, let qi denote the point with minimum height on the ith maximal underpass encountered as one traverses the knot from the same starting point.

Crossing Information Graph

• For any knot diagram, we define the crossing information graph (CIG) as follows:– There is a vertex for each pi and qi.

– Two vertices pi and qj are connected with an edge if the overpass containing pi crosses over the underpass containing qj.

Example of a CIG

Properties of a CIG

• Any ordered pair of connected vertices (pi, qj) will be called a pairing.

• A matching is a set of pairings with no common vertices.

• A diagram is said to be paired if its CIG contains a perfect matching.

Example of a Paired Diagram

Terminology

• Each knot diagram has a crossing number c (the number of crossings), and a bridge number n (the number of maximal overpasses).

c = 3n = 3

Reduced Knot Diagrams

• A knot diagram is said to be reduced if no crossings can be undone by untwisting part of the knot, as below.

T1 and T2 are knotted arcs.

Ropelength

• The ropelength of a conformation is defined as the ratio between the arc length of the conformation and its injectivity radius.– The injectivity radius is the maximum radius a tube

surrounding the knot can have without intersecting itself.

• Intuitively, ropelength is a measure of how much rope is needed to tie a knot.

• The ropelength of a knot type is the minimum of the ropelengths of all its conformations

Examples of Ropelength-Minimizing Conformations

Ropelength

• The only knot type whose minimal ropelength is known is the unknot (2π).

• Sadjadi (2007) and Alley (2009) have each found lower bounds on the ropelength of special classes of conformations.– Sadjadi: for alternating reduced diagrams, Rop(K) ≥ 4c.– Alley: for paired reduced diagrams, Rop(K) ≥ 4n.

• One goal of this study was to generalize these results to all reduced diagrams, and perhaps to all knot diagrams.

Ropelength of Reduced Knot Diagrams

• It has been shown that ropelength is scale invariant, so we will assume that the injectivity radius is 1. The ropelength will then be equal to the arc length of the conformation.

• Under this assumption, if we let oi be the height of pi and uj be the height of qj, then

oi - uj ≥ 2 for each pairing (pi, qj).

General Idea for Finding a Lower Bound for Ropelength

• We will attempt to minimize the vertical distance traveled as one traverses the knot; this will then be a lower bound for the ropelength.

• Because crossing strands must be separated by at least 2 units in height, there must be some vertical distance, but how much?

The Taut Diagram

• Given a conformation K, we define the taut image of K by projecting K onto the z-axis, and then replacing the arc between each vi and vi+1 with a straight line segment. The taut diagram t(K) is a graph of the taut image.

Example of t(K)

The Length of the Taut Diagram

• The sum of the vertical lengths of the line segments in t(K) can be thought of as the minimal vertical distance one has to travel when traversing the knot.

• Thus, given the constraint that the elements of pairings be at least two units apart, finding the minimum length of t(K) will give a lower bound for the ropelength of K.

The Reduced Taut Diagram

• Given a taut diagram t(K), we define the reduced taut diagram rt(K) by redefining the heights of v1, …, v2n (in that order) as follows:– Move each underpass down a minimal distance so

that it lies below its adjacent overpasses.– Move each overpass up a minimal distance so that

it lies above its adjacent underpasses.

Example of Defining rt(K)

t(K) rt(K)

The Utility of rt(K)

• Theorem: The length of rt(K) is less than or equal to the length of t(K).

• Thus, we can find the minimum length of rt(K) instead of t(K), as they are equal.

The Utility of rt(K)

• The length of rt(K) is equal to the sum of the vertical lengths of its line segments. But since we know that each overpass is above its adjacent underpasses, this equals:

(h(v2) – h(v1)) + (h(v2) – h(v3)) + …

… + (h(v2n) – h(v2n-1)) + (h(v2n) – h(v1)) =

=

The Utility of rt(K)

• Thus, the function we are trying to minimize is linear.

• In addition, it is only subject to linear inequality constraints requiring that: – Elements of pairings be at least two units apart– Overpasses be above their adjacent underpasses.

• Such a minimalization problem is called a linear programming (LP) problem.

Linear Programming

• More specifically, a linear programming problem is one which can be expressed as:– Minimize c · x, subject to:– Ax = b– x ≥ 0 where A is a constant matrix, and b and c are

constant vectors of appropriate dimensions.

The Simplex Algorithm

• The simplex algorithm allows us to obtain an optimal solution to any LP, provided the following requirements are met:– Rank(A) is maximal.– Ax = b is consistent.– {c · x | Ax = b and x ≥ 0} is bounded from below.

The Simplex Algorithm

• The simplex algorithm starts with a matrix (called a tableau) of the following form:

0th row

0th column

Example of the Simplex Algorithm

Applying the Simplex Algorithm

• The only thing preventing us from being able to use the simplex algorithm to find the minimum length of rt(K) is knowing which set of columns to choose at the beginning.– Fortunately, one can use the CIG to choose a set

of columns which will always produce a bfs.• Thus, we can always use this algorithm to find

the minimum length of rt(K), which is a lower bound for the ropelength of K.

Applying the Simplex Algorithm

• Using this approach, we can also re-prove the results of Sadjadi and Alley.

• In addition, we can use the theory behind the simplex algorithm to prove the following:

• Theorem: There exists an arrangement of rt(K) with minimum length in which each overpass and underpass has a height equal to an integer multiple of 2.

Theory Behind the Simplex Algorithm

• Any LP can also be expressed as the following:– Minimize c · x, subject to:– Ax ≤ b– x ≥ 0

• When written in this form, the set of all values that satisfy the above constraints forms a convex polytope.

Theory Behind the Simplex Algorithm

• The simplex algorithm relies on the fact that the function to be minimized takes its optimal value at one of the vertices of the polytope.

• One can prove that, if any of the values is not equal to an integer multiple of 2, it cannot be a vertex of the corresponding polytope.

Future Research

• Although we have an algorithm with which to find a lower bound for the ropelength of any reduced conformation, it is not yet known how (or if) this lower bound can be obtained directly through properties of the knot diagram and CIG.

• This would be useful, since applying the simplex algorithm here often results in working with unwieldy matrices.

Such as…

Future Research

• Our final theorem greatly decreases the number of arrangements of rt(K) we need to consider when minimizing its length. Can an alternate algorithm which is better tailored to this problem be developed?

• In order to consider all possible conformations of a knot, we still need to consider non-reduced diagrams.