L11n102

L11n101

L11n103

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n102 at Knotilus!

Link Presentations

[edit Notes on L11n102's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X13,18,14,19 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 8, -7, 9, -11, 2, -5, 6, -8, 3, -4, 5, -9, 7, -6, 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{(t(1)-1) (t(2)-1) \left(3 t(2)^2-4 t(2)+3\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{10}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^9 z^{-1} -2 a^7 z^3-a^7 z+a^7 z^{-1} +2 a^5 z^5+5 a^5 z^3+5 a^5 z+2 a^5 z^{-1} +a^3 z^5-3 a^3 z-2 a^3 z^{-1} -a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -6 z^4 a^{10}+11 z^2 a^{10}-4 a^{10}-3 z^7 a^9+3 z^5 a^9-z^3 a^9+2 z a^9+a^9 z^{-1} -5 z^8 a^8+13 z^6 a^8-24 z^4 a^8+25 z^2 a^8-9 a^8-2 z^9 a^7-5 z^7 a^7+20 z^5 a^7-25 z^3 a^7+9 z a^7+a^7 z^{-1} -10 z^8 a^6+23 z^6 a^6-24 z^4 a^6+12 z^2 a^6-4 a^6-2 z^9 a^5-7 z^7 a^5+27 z^5 a^5-33 z^3 a^5+15 z a^5-2 a^5 z^{-1} -5 z^8 a^4+7 z^6 a^4-z^4 a^4-3 z^2 a^4+2 a^4-5 z^7 a^3+9 z^5 a^3-7 z^3 a^3+7 z a^3-2 a^3 z^{-1} -3 z^6 a^2+5 z^4 a^2-z^2 a^2-z^5 a+2 z^3 a-z a }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 2   -2 -2               5 1   4 -4             7 4     -3 -6           7 3       4 -8         6 7         1 -10       7 7           0 -12     3 6             3 -14   3 7               -4 -16   3                 3 -18 3                   -3

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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}_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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