L11n80

L11n79

L11n81

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n80 at Knotilus!

Link Presentations

[edit Notes on L11n80's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X19,1,20,4 X5,14,6,15 X3,10,4,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -8, 10, -9, 6, 11, -2, 4, -3, -5, 8, -10, 9}

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+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{7}{q^{17/2}}+\frac{5}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^9-4 z^3 a^9-6 z a^9-3 a^9 z^{-1} +z^7 a^7+6 z^5 a^7+15 z^3 a^7+18 z a^7+7 a^7 z^{-1} -2 z^5 a^5-9 z^3 a^5-12 z a^5-4 a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-2 a^{14} z^2+a^{14}+2 a^{13} z^5-2 a^{13} z^3+3 a^{12} z^6-3 a^{12} z^4+a^{12} z^2+3 a^{11} z^7-3 a^{11} z^5+2 a^{11} z^3+2 a^{10} z^8-a^{10} z^6+a^{10} z^2+a^9 z^9-a^9 z^7+6 a^9 z^5-11 a^9 z^3+8 a^9 z-3 a^9 z^{-1} +3 a^8 z^8-8 a^8 z^6+15 a^8 z^4-19 a^8 z^2+7 a^8+a^7 z^9-4 a^7 z^7+14 a^7 z^5-27 a^7 z^3+22 a^7 z-7 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+11 a^6 z^4-17 a^6 z^2+7 a^6+3 a^5 z^5-12 a^5 z^3+14 a^5 z-4 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   2 2 -6                 2 2 0 -8               4     4 -10             3 2     -1 -12           5 4       1 -14         3 3         0 -16       4 5           -1 -18     1 3             2 -20   1 4               -3 -22   1                 1 -24 1                   -1

Integral Khovanov Homology (db, data source)

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