L11a19

L11a18

L11a20

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Link Presentations

[edit Notes on L11a19's Link Presentations]

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

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(t(2)^4-4 t(2)^3+4 t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 16 q^{9/2}-18 q^{7/2}+17 q^{5/2}-\frac{1}{q^{5/2}}-16 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+11 \sqrt{q}-\frac{7}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -7 z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +9 z a^{-3} +3 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-6 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -8 z^9 a^{-3} -5 z^9 a^{-5} -10 z^8 a^{-2} -16 z^8 a^{-4} -9 z^8 a^{-6} -3 z^8-a z^7+4 z^7 a^{-1} +10 z^7 a^{-3} -4 z^7 a^{-5} -9 z^7 a^{-7} +41 z^6 a^{-2} +49 z^6 a^{-4} +13 z^6 a^{-6} -6 z^6 a^{-8} +11 z^6+4 a z^5+13 z^5 a^{-1} +25 z^5 a^{-3} +33 z^5 a^{-5} +14 z^5 a^{-7} -3 z^5 a^{-9} -42 z^4 a^{-2} -40 z^4 a^{-4} -4 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} -13 z^4-6 a z^3-25 z^3 a^{-1} -40 z^3 a^{-3} -36 z^3 a^{-5} -13 z^3 a^{-7} +2 z^3 a^{-9} +15 z^2 a^{-2} +11 z^2 a^{-4} -4 z^2 a^{-6} -4 z^2 a^{-8} +z^2 a^{-10} +5 z^2+4 a z+13 z a^{-1} +20 z a^{-3} +16 z a^{-5} +5 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} 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 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 7 2     5 10               9 5       -4 8             9 7         2 6           8 9           1 4         8 9             -1 2       5 10               5 0     2 6                 -4 -2   1 5                   4 -4   2                     -2 -6 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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