L11a446

L11a445

L11a447

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Link Presentations

[edit Notes on L11a446's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X18,9,19,10 X16,8,17,7 X22,17,13,18 X20,11,21,12 X8,14,9,13 X10,19,11,20 X12,21,5,22 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 3, -8, 6, -9}, {7, -2, 11, -4, 5, -3, 8, -6, 9, -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{u v^3 w^3-u v^3 w^2-2 u v^2 w^3+4 u v^2 w^2-2 u v^2 w+u v w^3-3 u v w^2+4 u v w-u v+u w^2-2 u w+u-v^3 w^3+2 v^3 w^2-v^3 w+v^2 w^3-4 v^2 w^2+3 v^2 w-v^2+2 v w^2-4 v w+2 v+w-1}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-3 q^2+6 q-8+13 q^{-1} -14 q^{-2} +15 q^{-3} -12 q^{-4} +10 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-14 a^2 z^4+4 z^4-3 a^6 z^2+13 a^4 z^2-16 a^2 z^2+5 z^2-2 a^6+7 a^4-9 a^2+4+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-6 a^8 z^4+3 a^8 z^2-a^8+4 a^7 z^7-5 a^7 z^5-2 a^7 z^3+a^7 z+4 a^6 z^8-4 a^6 z^6-2 a^6 z^4+a^6 z^2+3 a^5 z^9-2 a^5 z^7-3 a^5 z^5+6 a^5 z^3-3 a^5 z+a^4 z^{10}+7 a^4 z^8-29 a^4 z^6+47 a^4 z^4-31 a^4 z^2-a^4 z^{-2} +9 a^4+7 a^3 z^9-20 a^3 z^7+19 a^3 z^5+4 a^3 z^3-8 a^3 z+2 a^3 z^{-1} +a^2 z^{10}+8 a^2 z^8-41 a^2 z^6+z^6 a^{-2} +67 a^2 z^4-3 z^4 a^{-2} -44 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+4 a z^9-11 a z^7+3 z^7 a^{-1} +7 a z^5-9 z^5 a^{-1} +2 a z^3+4 z^3 a^{-1} -5 a z+2 a z^{-1} +5 z^8-18 z^6+21 z^4-14 z^2- z^{-2} +6 }[/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 7                       1 1 5                     2   -2 3                   4 1   3 1                 4 2     -2 -1               9 4       5 -3             8 7         -1 -5           7 6           1 -7         5 8             3 -9       5 7               -2 -11     2 6                 4 -13   1 4                   -3 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

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