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

[Jonathan Post]

Yes, I think so, Franklin T. Adams-Watters.

The paper seems to set some upper and lower bounds, which I am not
100% sure about.

Would you be willing to co-submit this with me?  I'm having a severe
LAN problem at my home office, even after rebooting multiple PCs,
cable modem, router? Hence I also apologize for shouting fire in a
crowded seqfans theatre about wondering if OEIS was down.  After
working with software/hardware systems since 1966, you'd think I'd run
end-to-end checks more often...

On the plus side, it gets me away from Facebook.

Best,

Jonathan Vos Post

On Mon, Aug 17, 2009 at 11:53 AM, <franktaw at netscape.net> wrote:
> 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,...?
>
> Franklin T. Adams-Watters
>
>
> -----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
>
>
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