L11n177

L11n176

L11n178

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n177 at Knotilus!

Link Presentations

[edit Notes on L11n177's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X5,15,6,14 X18,12,19,11 X10,4,11,3 X7,13,8,12 X13,17,14,16 X17,7,18,22 X15,1,16,6 X4,21,5,22 X2,20,3,19
Gauss code	{1, -11, 5, -10, -3, 9}, {-6, -1, 2, -5, 4, 6, -7, 3, -9, 7, -8, -4, 11, -2, 10, 8}

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^2 v^4-2 u^2 v^3+3 u^2 v^2-3 u^2 v+u^2-2 u v^4+5 u v^3-5 u v^2+5 u v-2 u+v^4-3 v^3+3 v^2-2 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{15/2}+6 q^{13/2}-9 q^{11/2}+12 q^{9/2}-14 q^{7/2}+12 q^{5/2}-11 q^{3/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{-7} -z^5 a^{-5} -z^3 a^{-5} +z a^{-5} +z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} +z a^{-3} - a^{-3} z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^3 a^{-9} -z a^{-9} +z^6 a^{-8} +6 z^4 a^{-8} -5 z^2 a^{-8} +4 z^7 a^{-7} -3 z^5 a^{-7} +4 z^3 a^{-7} -z a^{-7} +5 z^8 a^{-6} -8 z^6 a^{-6} +10 z^4 a^{-6} -6 z^2 a^{-6} +2 z^9 a^{-5} +6 z^7 a^{-5} -19 z^5 a^{-5} +12 z^3 a^{-5} -z a^{-5} +10 z^8 a^{-4} -21 z^6 a^{-4} +8 z^4 a^{-4} +2 z^9 a^{-3} +6 z^7 a^{-3} -27 z^5 a^{-3} +19 z^3 a^{-3} -2 z a^{-3} - a^{-3} z^{-1} +5 z^8 a^{-2} -11 z^6 a^{-2} +2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -11 z^5 a^{-1} +8 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +z^6-2 z^4+z^2 }[/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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 16                   2 2 14                 4   -4 12               5 2   3 10             7 4     -3 8           7 5       2 6         6 8         2 4       5 6           -1 2     3 7             4 0   1 4               -3 -2   3                 3 -4 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=-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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

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