L11n86

L11n85

L11n87

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n86 at Knotilus!

Link Presentations

[edit Notes on L11n86's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X18,10,19,9 X19,5,20,22 X15,21,16,20 X21,17,22,16 X8,18,9,17 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 11, -2, 9, -3, -6, 7, 8, -4, -5, 6, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{u v^5-2 u v^4+u v^3-u v^2+2 u v-2 u-2 v^5+2 v^4-v^3+v^2-2 v+1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{13/2}-3 q^{11/2}+5 q^{9/2}-5 q^{7/2}+6 q^{5/2}-6 q^{3/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} -z^5 a^{-5} -4 z^3 a^{-5} -3 z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +8 z^3 a^{-3} +7 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-9 z^3 a^{-1} +4 a z-11 z a^{-1} +3 a z^{-1} -5 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -4 z^8 a^{-2} -3 z^8 a^{-4} -z^8-a z^7+2 z^7 a^{-1} -3 z^7 a^{-5} +15 z^6 a^{-2} +11 z^6 a^{-4} -z^6 a^{-6} +3 z^6+6 a z^5+6 z^5 a^{-1} +9 z^5 a^{-3} +9 z^5 a^{-5} -11 z^4 a^{-2} -14 z^4 a^{-4} -z^4 a^{-6} +2 z^4-12 a z^3-15 z^3 a^{-1} -9 z^3 a^{-3} -9 z^3 a^{-5} -3 z^3 a^{-7} -4 z^2 a^{-2} +8 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} -9 z^2+10 a z+13 z a^{-1} +4 z a^{-3} +2 z a^{-5} +z a^{-7} +5 a^{-2} - a^{-6} +5-3 a z^{-1} -5 a^{-1} z^{-1} -2 a^{-3} z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 14                   1 -1 12                 2   2 10               3 1   -2 8             2 2     0 6           4 3       -1 4         2 2         0 2       3 5           2 0     1 1             0 -2     3               3 -4 1 1                 0 -6 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L11n85 L11n87