Knots On a Cubic Lattice

Let the function f from S^1 to R^3 be defined as follows.

f(2 pi k / N) = (m_k, n_k, l_k)
|m_{k+1}-m_k|+|n_{k+1}-n_k|+|l_{k+1}-l_k|=1
f(2 pi (k+r) / N) = r (m_k, n_k, l_k) + (1-r) (m_{k+1}, n_{k+1}, l_{k+1})

where N is a constant, k is an integer between 0 and N, 0 < r < 1, while m_k,n_k,l_k are integers. Such a function defines a knot if and only if (m_0, n_0, l_0) = (m_N, n_N, l_N) while for any i,j not equal to each other, other than 0 and N, (m_i, n_i, l_i) is not equal to (m_j, n_j, l_j). The first condition implies that N must be even.

Such a function may be denoted as follows. First, one observes that the vectors (m_{k+1}-m_k, n_{k+1}-n_k, l_{k+1}-n_k) are unit vectors and may be along one of the six axes x,y,z,-x,-y,-z. Once these are defined, the function f and thus the knot are fixed. Therefore for a value of N there are at most 6^N such knots. In reality, due to the constraints shown above, this number is much smaller. These knots are denoted through a sequence a_1, a_2, ..., a_N, where a_k takes values in {1,2,3,4,5,6}, and a_k=1,2,3,4,5,6 means that (m_{k+1-m_k, n_{k+1}-n_k, l_{k+1}-l_k) is along the axes x,y,z,-z,-y,-x respectively.

As was the case with the notation based on regular projections, the following questions emerge.

• Which notations correspond to actual knots?
• If two knots are ambient isotopic, how are the notations related?
• If two knots are inequivalent, how may one distinguish them through their notations?

To proceed with answers to these questions, click here

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Educational institutions are encouraged to reproduce and distribute these materials for educational use free of charge as long as credit and notification are provided. For any other purpose except educational, such as commercial etc, use of these materials is prohibited without prior written permission.

Charilaos Aneziris, aneziris@hades.ifh.de

Copyright 1996