L11a82

L11a81

L11a83

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a82 at Knotilus!

Link Presentations

[edit Notes on L11a82's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X22,13,5,14 X14,17,15,18 X20,10,21,9 X18,21,19,22 X8,16,9,15 X10,20,11,19 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 6, -9, 11, -2, 4, -5, 8, -3, 5, -7, 9, -6, 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{2 u v^4-10 u v^3+14 u v^2-6 u v+u+v^5-6 v^4+14 v^3-10 v^2+2 v}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{14}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+3 q^{5/2}-\frac{22}{q^{5/2}}-8 q^{3/2}+\frac{21}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+14 \sqrt{q}-\frac{18}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+3 z^3 a^5+3 z a^5+2 a^5 z^{-1} -2 z^5 a^3-4 z^3 a^3-8 z a^3-4 a^3 z^{-1} -z^5 a+3 z^3 a+6 z a+3 a z^{-1} +2 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} -z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-10 a^7 z^5+8 a^7 z^3-3 a^7 z+6 a^6 z^8-12 a^6 z^6+5 a^6 z^4-a^6 z^2+a^6+4 a^5 z^9+5 a^5 z^7-34 a^5 z^5+36 a^5 z^3-15 a^5 z+2 a^5 z^{-1} +a^4 z^{10}+17 a^4 z^8-47 a^4 z^6+40 a^4 z^4-14 a^4 z^2+2 a^4+9 a^3 z^9-a^3 z^7-40 a^3 z^5+z^5 a^{-3} +53 a^3 z^3-2 z^3 a^{-3} -24 a^3 z+z a^{-3} +4 a^3 z^{-1} +a^2 z^{10}+19 a^2 z^8-51 a^2 z^6+3 z^6 a^{-2} +53 a^2 z^4-4 z^4 a^{-2} -24 a^2 z^2+z^2 a^{-2} +3 a^2+5 a z^9+4 a z^7+6 z^7 a^{-1} -26 a z^5-9 z^5 a^{-1} +34 a z^3+7 z^3 a^{-1} -17 a z-4 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +8 z^8-14 z^6+16 z^4-11 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   6 1   5 2                 8 2     -6 0               10 6       4 -2             12 9         -3 -4           10 9           1 -6         8 12             4 -8       6 10               -4 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L11a81 L11a83