L11n251

L11n250

L11n252

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n251 at Knotilus!

Link Presentations

[edit Notes on L11n251's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X8394 X11,17,12,16 X14,8,15,7 X6,16,7,15 X17,11,18,22 X4,20,5,19 X18,6,19,5 X20,9,21,10 X10,21,1,22
Gauss code	{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {-4, -1, 2, -5, 6, 4, -7, -9, 8, -10, 11, 7}

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{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-t(2) t(1)^2+t(1)^2+3 t(2) t(1)+t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{9/2}}-2 q^{7/2}+\frac{7}{q^{7/2}}+5 q^{5/2}-\frac{11}{q^{5/2}}-8 q^{3/2}+\frac{11}{q^{3/2}}+\frac{1}{q^{11/2}}+11 \sqrt{q}-\frac{13}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-3 a^3 z^3-4 a^3 z-z a^{-3} +a z^7+5 a z^5-z^5 a^{-1} +10 a z^3-2 z^3 a^{-1} +7 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4-a^6 z^2+3 a^5 z^5-2 a^5 z^3+6 a^4 z^6-7 a^4 z^4+4 a^4 z^2+8 a^3 z^7-15 a^3 z^5+3 z^5 a^{-3} +16 a^3 z^3-6 z^3 a^{-3} -8 a^3 z+2 z a^{-3} +5 a^2 z^8+z^8 a^{-2} -3 a^2 z^6+2 z^6 a^{-2} -6 a^2 z^4-9 z^4 a^{-2} +7 a^2 z^2+6 z^2 a^{-2} +a z^9+z^9 a^{-1} +11 a z^7+3 z^7 a^{-1} -30 a z^5-9 z^5 a^{-1} +29 a z^3+5 z^3 a^{-1} -14 a z-4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +6 z^8-7 z^6-7 z^4+8 z^2-1 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   2 2 6                 3   -3 4               5 2   3 2             6 3     -3 0           7 5       2 -2         6 8         2 -4       5 5           0 -6     2 6             4 -8   1 5               -4 -10   2                 2 -12 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=0 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}_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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