L11n88

L11n87

L11n89

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n88 at Knotilus!

Link Presentations

[edit Notes on L11n88's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X9,15,10,14 X19,10,20,11 X21,9,22,8 X13,18,14,19 X15,21,16,20 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 7, -5, 6, -11, 2, -8, 5, -9, 3, -4, 8, -6, 9, -7, 4}

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-1) (v-1)^3 \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{8}{q^{7/2}}+8 q^{5/2}-\frac{12}{q^{5/2}}-13 q^{3/2}+\frac{15}{q^{3/2}}+15 \sqrt{q}-\frac{17}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-a^3 z^5+4 a z^5-2 z^5 a^{-1} -3 a^3 z^3+7 a z^3-5 z^3 a^{-1} +z^3 a^{-3} +a^5 z-5 a^3 z+7 a z-4 z a^{-1} +z a^{-3} +a^5 z^{-1} -3 a^3 z^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 6 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-2 z^4 a^{-4} -7 a^4 z^2+z^2 a^{-4} +a^4+10 a^3 z^7+4 z^7 a^{-3} -21 a^3 z^5-10 z^5 a^{-3} +28 a^3 z^3+8 z^3 a^{-3} -16 a^3 z-3 z a^{-3} +3 a^3 z^{-1} +10 a^2 z^8+6 z^8 a^{-2} -22 a^2 z^6-13 z^6 a^{-2} +27 a^2 z^4+5 z^4 a^{-2} -17 a^2 z^2+3 a^2+ a^{-2} +3 a z^9+3 z^9 a^{-1} +13 a z^7+7 z^7 a^{-1} -48 a z^5-37 z^5 a^{-1} +49 a z^3+35 z^3 a^{-1} -22 a z-13 z a^{-1} +4 a z^{-1} +2 a^{-1} z^{-1} +16 z^8-39 z^6+28 z^4-11 z^2+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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 3   3 6               5 1   -4 4             8 3     5 2           7 5       -2 0         10 8         2 -2       7 9           2 -4     5 8             -3 -6   3 7               4 -8   5                 -5 -10 3                   3

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

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