# [seqfan] Re: Number of knots with trivializing number k among all knots with n crossings

franktaw at netscape.net franktaw at netscape.net
Mon Aug 17 20:53:48 CEST 2009

```Yes, this is definitely a good sequence for the OEIS.

Yes, as I understand it, odd numbered columns should be eliminated.  So
the definition becomes something like "Table T(n,k) is the number of
knots with n crossings and trivializing number 2k".  This will then be
a tabf sequence:
1,1,0,1,1,2,1,5,6,1,11,9,...

Yes, this is the way to present it.

Do we generally have 2k <= n - 3, so the row lengths are
1,1,2,2,3,3,4,4,5,5,6,6,...?

-----Original Message-----
From: Jonathan Post <jvospost3 at gmail.com>

Number of knots with trivializing number k among all knots with n
crossings.

derived from Table 1, p.24, of
Allison Henrich, Noel MacNaughton, Sneha Narayan, Oliver Pechenik,
Jennifer Townsend,
Classical and Virtual Pseudodiagram Theory and New Bounds on
Unknotting Numbers and Genus
http://arxiv.org/PS_cache/arxiv/pdf/0908/0908.1981v1.pdf

n / k=1.|.k=2.|.k=3.|.k=4.|.k=5.|.k=6.|.
n=3.|.0. |.1.|
n=4.|.0. |.1.|
n=5.|.0.|.0.|.0.|..1.|
n=6.|.0.|.1.|.0.|..2.|
n=7.|.0.|.1.|.0.|..5.|.0.|.6.|
n=8.|.0.|.1.|.0.|.11.|.0.|.9.|

The n=9 row of the table seems currently uncertain for some values of k.

Is this good for OEIS? Should one eliminate the non-even k columns,
per theorem in the paper? Is there a better presentation?

Thank you,

Jonathan Vos Post

```