On an alternative stratification of knots

Abstract

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

arXiv comment12 pages

References in corpus (15)

• On Khovanov's categorification of the Jones polynomial
• On Link Invariants and Topological String Amplitudes
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links III : Compact Semi-simple Group
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links
• Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Colored HOMFLY polynomials for the pretzel knots and links
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Chirality of Knots $9_{42}$ and $10_{71}$ and Chern-Simons Theory
• Knot invariants from Virasoro related representation and pretzel knots
• Colored Kauffman Homology and Super-A-polynomials
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• Universal Knot Diagrams
• Exchange Relation in sl3 WZNW model in Semiclassical Limit