Research & Projects

The history of knot tabulation stretches back almost 200 years. As of now prime knots up to reversal and mirror imaging have been successfully classified up to crossing number 20 using knots invariants like the Jones Polynomial.

Alternating knots have been classified up to crossing number 23.

The Quandle State-Sum Invariant was first introduced in 1999 and has been used to recreate the classical of knots up to crossing number 13 as of 2024.

Objectives

1. Classify knots of crossing number 14 using the Quandle State-Sum Invariant.

2. Classify all knots up to crossing number 13 via the State-Sum Invariant using fewer than 24 quandles (which is the current fewest number used).

3. (Long Term Goal) Classify all knots up to crossing number 23 using the Quandle State-Sum Invariant.

Results

In progress.

1. It is possible to create a new quandle given two other quandles.

2. If you can distinguish all knots up to crossing number n using any enhancement of the Quandle Counting Invariant, then there are infinite such quandles.

Legendrian Knots can sometimes be fully classified using their topological knot type, tb number and rotation number. When this is the case, we call them Legendrian Simple knots. However, thanks to Chekanov, we know knots like m(5_2) are non-simple.

We call invariants that can distinguish non-simple Legendrian knots, effective.

In 2019 Ceniceros, Elhamdadi, and Nelson defined Legendrian Racks, a counting invariant for them, and enumerated them through order 4.

Objectives

1. Enumerate Legendrian Racks up to order 10.

2. Show the associated Legendrian Rack counting invariant is effective at distinguishing the Chekanov pair, m(5_2).

3. Fully distinguish 6_2 and other open Legendrian knot questions.

Results

In progress.

1. Have enumerated over one million Legendrian racks

Some quandles have what are called "Good Involutions". Not a lot is known about which quandles have this feature. There are also many other algebraic properties of quandles still uncomputed.

Objectives

1. Enumerate all good involutions of quandles with order less than or equal to 6.

2. Calculate Inner Automorphism, Automorphism, and Transvection groups of quandles up to order 8, 9, and 10

3. Determine the Transvection groups of trivial, dihedral, Alexander, latin, medial, connected, symmetric, and other classes of quandles.

4. Determine the Inner and Automorphism groups of latin, medial, connected, symmetric, and other classes of quandles.

Results

In progress.

1. Computation of Inn, Aut half completed for quandles of order 8

2. Computation of Transv completed for quandles of less than or equal to order 4.

3. Proved Transvection groups of trivial and dihedral quandles

Another type of knot is called Transverse. Quandle theorists have yet to tackle these objects.

Objectives

1. Prove a minimal generating set of oriented Transverse Reidemeister moves.

2. Define a quandle-like structure on Transverse Knots.

3. Distinguish several Transverse knots using counting invariant

Results

In progress.