L11a288

L11a287

L11a289

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a288 at Knotilus!

Link Presentations

[edit Notes on L11a288's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X16,9,17,10 X20,12,21,11 X22,15,9,16 X14,6,15,5 X4,14,5,13 X6,20,7,19 X18,8,19,7 X8,18,1,17 X2,21,3,22
Gauss code	{1, -11, 2, -7, 6, -8, 9, -10}, {3, -1, 4, -2, 7, -6, 5, -3, 10, -9, 8, -4, 11, -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) \left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{15/2}-3 q^{13/2}+6 q^{11/2}-11 q^{9/2}+13 q^{7/2}-16 q^{5/2}+16 q^{3/2}-14 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-5} +3 z^3 a^{-5} +2 z a^{-5} - a^{-5} z^{-1} -z^7 a^{-3} -4 z^5 a^{-3} -5 z^3 a^{-3} -z a^{-3} + a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +3 a z^3-5 z^3 a^{-1} +2 a z-z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^{10} a^{-2} -2 z^{10} a^{-4} -5 z^9 a^{-1} -9 z^9 a^{-3} -4 z^9 a^{-5} -2 z^8 a^{-2} -4 z^8 a^{-6} -6 z^8-5 a z^7+12 z^7 a^{-1} +29 z^7 a^{-3} +9 z^7 a^{-5} -3 z^7 a^{-7} -3 a^2 z^6+16 z^6 a^{-2} +8 z^6 a^{-4} +9 z^6 a^{-6} -z^6 a^{-8} +15 z^6-a^3 z^5+11 a z^5-13 z^5 a^{-1} -40 z^5 a^{-3} -6 z^5 a^{-5} +9 z^5 a^{-7} +6 a^2 z^4-26 z^4 a^{-2} -9 z^4 a^{-4} -2 z^4 a^{-6} +3 z^4 a^{-8} -16 z^4+2 a^3 z^3-7 a z^3+2 z^3 a^{-1} +21 z^3 a^{-3} +3 z^3 a^{-5} -7 z^3 a^{-7} -a^2 z^2+9 z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} -2 z^2 a^{-8} +6 z^2+2 a z-z a^{-1} -2 z a^{-3} +3 z a^{-5} +2 z a^{-7} + a^{-4} - a^{-3} z^{-1} - a^{-5} 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 6 7 χ 16                       1 -1 14                     2   2 12                   4 1   -3 10                 7 2     5 8               7 5       -2 6             9 6         3 4           7 7           0 2         7 9             -2 0       4 8               4 -2     2 6                 -4 -4   1 4                   3 -6   2                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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.

L11a287 L11a289