L11n87

L11n86

L11n88

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n87 at Knotilus!

Link Presentations

[edit Notes on L11n87's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X15,20,16,21 X9,19,10,18 X19,9,20,8 X17,22,18,5 X21,16,22,17 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -9, 11, -2, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7}

A Braid Representative	{{{braid_table}}}
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^3-u v^2-u-v^3-v+1}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{7/2}}-q^{5/2}+\frac{1}{q^{5/2}}+2 q^{3/2}-\frac{2}{q^{3/2}}-2 \sqrt{q}+\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-1} -a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^5+4 a z^3-z^3 a^{-1} +3 a z-2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-a^4 z^8-3 a^2 z^8-2 z^8+5 a^3 z^7+4 a z^7-z^7 a^{-1} +6 a^4 z^6+17 a^2 z^6+11 z^6-5 a^3 z^5+5 z^5 a^{-1} -9 a^4 z^4-25 a^2 z^4-16 z^4+a^5 z^3+a^3 z^3-6 a z^3-6 z^3 a^{-1} +a^6 z^2+4 a^4 z^2+10 a^2 z^2+7 z^2+a^5 z+a z+2 z a^{-1} +a^4-a^5 z^{-1} -a^3 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 χ 6                 1 1 4               1   -1 2             1 1   0 0         1 2 1     0 -2       1 2 1       0 -4       1 2         1 -6     2 1           1 -8 1   1             2 -10 1 1               0

Integral Khovanov Homology (db, data source)

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