OFFSET
0,2
COMMENTS
From Andrey Zabolotskiy, Dec 24 2018: (Start)
It follows from Definition 7 of the paper by Cantarella, Chapman & Mastin that every one of A008988(n) knot shadows contributes some diagrams to a(n). The number of diagrams contributed is shown near knot shadows in the illustration linked below (e.g., the knot shadow looking like an n-gon with petals contributes A000031(n) inequivalent diagrams, while non-symmetric knot shadows contribute 2^n diagrams each). Note that the orientation of knot shadows is not shown when it is not needed to distinguish two knot shadows, but all the knot shadows are oriented, which spoils the symmetry of many of them. (Even though the paper itself starts from unoriented knot shadows counted by A008989, eventually the orientation is assigned to the diagrams.)
The more symmetric a knot shadow is, the fewer inequivalent diagrams it contributes. The knot shadows and the diagrams generated by them are oriented but immersed into unoriented spheres. When determining equivalent diagrams, they can be rotated on the sphere and turned "inside out" but not reflected. Some planar drawings of knot shadows make them look less symmetric than they actually are (taking into account their immersion into the sphere).
Note that this equivalence relation distinguishes many more knot diagrams than the ambient isotopy of knots, cf. A086825. (End)
LINKS
J. Cantarella, H. Chapman, M. Mastin, Knot Probabilities in Random Diagrams, arXiv preprint arXiv:1512.05749 [math.GT], 2015. See Tables I and III.
Andrey Zabolotskiy, Illustration of the knot shadows contributing to a(0)-a(4)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Nov 07 2016
EXTENSIONS
a(0)-a(2) from Andrey Zabolotskiy, Dec 24 2018
STATUS
approved