L11a83

L11a82

L11a84

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Link Presentations

[edit Notes on L11a83's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,8,17,7 X20,10,21,9 X22,18,5,17 X18,22,19,21 X8,20,9,19 X14,12,15,11 X10,16,11,15 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, 4, -9, 8, -2, 11, -8, 9, -3, 5, -6, 7, -4, 6, -5}

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(4 t(2)^2-5 t(2)+4\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{21/2}-4 q^{19/2}+8 q^{17/2}-11 q^{15/2}+15 q^{13/2}-17 q^{11/2}+16 q^{9/2}-14 q^{7/2}+9 q^{5/2}-6 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-9} -z^5 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -4 z^3 a^{-5} -4 z a^{-5} -2 a^{-5} z^{-1} -z^5 a^{-3} -z^3 a^{-3} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -10 z^5 a^{-11} +5 z^3 a^{-11} +6 z^8 a^{-10} -15 z^6 a^{-10} +8 z^4 a^{-10} +4 z^9 a^{-9} -4 z^7 a^{-9} -6 z^5 a^{-9} +3 z^3 a^{-9} +z^{10} a^{-8} +8 z^8 a^{-8} -19 z^6 a^{-8} +5 z^4 a^{-8} +6 z^2 a^{-8} -2 a^{-8} +6 z^9 a^{-7} -6 z^7 a^{-7} -6 z^5 a^{-7} +10 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +z^{10} a^{-6} +5 z^8 a^{-6} -5 z^6 a^{-6} -7 z^4 a^{-6} +13 z^2 a^{-6} -5 a^{-6} +2 z^9 a^{-5} +5 z^7 a^{-5} -14 z^5 a^{-5} +15 z^3 a^{-5} -7 z a^{-5} +2 a^{-5} z^{-1} +3 z^8 a^{-4} -5 z^4 a^{-4} +6 z^2 a^{-4} -3 a^{-4} +3 z^7 a^{-3} -3 z^5 a^{-3} +2 z^6 a^{-2} -3 z^4 a^{-2} + a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +3 z a^{-1} - a^{-1} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     3   3 18                   5 1   -4 16                 6 3     3 14               9 5       -4 12             8 6         2 10           8 9           1 8         6 8             -2 6       3 8               5 4     3 6                 -3 2   1 5                   4 0   1                     -1 -2 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=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/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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