L11n318

L11n317

L11n319

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Link Presentations

[edit Notes on L11n318's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X20,11,21,12 X22,17,11,18 X16,21,17,22 X10,13,5,14 X19,8,20,9 X7,18,8,19 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, -9, 8, 10, -7}, {4, -2, 7, -3, 11, -6, 5, 9, -8, -4, 6, -5}

A Braid Representative	{{{braid_table}}}
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^3-2 u v^2 w^2+u v^2 w+u v w^4-3 u v w^3+5 u v w^2-3 u v w+u w^3-2 u w^2+2 u w-2 v^2 w^3+2 v^2 w^2-v^2 w+3 v w^3-5 v w^2+3 v w-v-w^3+2 w^2-w}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-11} +4 q^{-10} -8 q^{-9} +11 q^{-8} -14 q^{-7} +15 q^{-6} -12 q^{-5} +11 q^{-4} -5 q^{-3} +3 q^{-2} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{10}-a^{10} z^{-2} -a^{10}+3 z^4 a^8+8 z^2 a^8+4 a^8 z^{-2} +8 a^8-2 z^6 a^6-9 z^4 a^6-17 z^2 a^6-5 a^6 z^{-2} -15 a^6+3 z^4 a^4+9 z^2 a^4+2 a^4 z^{-2} +8 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{13} z^5-a^{13} z^3+4 a^{12} z^6-6 a^{12} z^4+a^{12} z^2+7 a^{11} z^7-13 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +6 a^{10} z^8-8 a^{10} z^6+2 a^{10} z^4-3 a^{10} z^2-a^{10} z^{-2} +2 a^{10}+2 a^9 z^9+9 a^9 z^7-28 a^9 z^5+27 a^9 z^3-13 a^9 z+5 a^9 z^{-1} +11 a^8 z^8-27 a^8 z^6+35 a^8 z^4-27 a^8 z^2-4 a^8 z^{-2} +13 a^8+2 a^7 z^9+5 a^7 z^7-20 a^7 z^5+33 a^7 z^3-27 a^7 z+9 a^7 z^{-1} +5 a^6 z^8-15 a^6 z^6+33 a^6 z^4-37 a^6 z^2-5 a^6 z^{-2} +20 a^6+3 a^5 z^7-6 a^5 z^5+13 a^5 z^3-16 a^5 z+5 a^5 z^{-1} +6 a^4 z^4-14 a^4 z^2-2 a^4 z^{-2} +10 a^4 }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   3 3 -5                 4 2 -2 -7               7 1   6 -9             6 5     -1 -11           9 6       3 -13         5 6         1 -15       6 9           -3 -17     3 6             3 -19   1 5               -4 -21   3                 3 -23 1                   -1

Integral Khovanov Homology (db, data source)

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