ON AN ALTERNATIVE STRATIFICATION OF KNOTS

摘要

We introduce an alternative stratification of knots: by the size of the lattice on which a knot can be first met. Using this classification, we find the fraction of unknots and knots with more than 10 minimal crossings inside different lattices and answer the question of which knots can be realized inside 3 x 3 and 5 x 5 lattices. In accordance with previous research, the fraction of unknots decreases exponentially with the growth of the lattice size. Our computational results are consistent with theoretical estimates for the number of knots with a fixed crossing number inside lattices of a given size.