L11a403

L11a402

L11a404

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Link Presentations

[edit Notes on L11a403's Link Presentations]

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

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^2 w^4-2 u v^2 w^3+2 u v^2 w^2-u v^2 w-2 u v w^4+5 u v w^3-5 u v w^2+4 u v w-u v+u w^4-3 u w^3+4 u w^2-3 u w+u-v^2 w^4+3 v^2 w^3-4 v^2 w^2+3 v^2 w-v^2+v w^4-4 v w^3+5 v w^2-5 v w+2 v+w^3-2 w^2+2 w-1}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-4 q^6+9 q^5-15 q^4+20 q^3-22 q^2+23 q-18+15 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-4} +3 z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} -z^8 a^{-2} -5 z^6 a^{-2} -a^2 z^4-10 z^4 a^{-2} -2 a^2 z^2-9 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^{-2} +2 z^6+7 z^4+7 z^2-2 z^{-2} +1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+5 a z^9+14 z^9 a^{-1} +9 z^9 a^{-3} +4 a^2 z^8+23 z^8 a^{-2} +15 z^8 a^{-4} +12 z^8+a^3 z^7-11 a z^7-28 z^7 a^{-1} -2 z^7 a^{-3} +14 z^7 a^{-5} -14 a^2 z^6-77 z^6 a^{-2} -25 z^6 a^{-4} +9 z^6 a^{-6} -57 z^6-3 a^3 z^5-2 a z^5-6 z^5 a^{-1} -30 z^5 a^{-3} -19 z^5 a^{-5} +4 z^5 a^{-7} +17 a^2 z^4+73 z^4 a^{-2} +15 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +67 z^4+3 a^3 z^3+13 a z^3+26 z^3 a^{-1} +27 z^3 a^{-3} +10 z^3 a^{-5} -z^3 a^{-7} -8 a^2 z^2-30 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -29 z^2-a^3 z-3 a z-7 z a^{-1} -7 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   6 1   5 9                 9 3     -6 7               11 6       5 5             12 10         -2 3           11 10           1 1         9 14             5 -1       6 9               -3 -3     3 10                 7 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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