L11n276

L11n275

L11n277

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Link Presentations

[edit Notes on L11n276's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X22,17,11,18 X20,11,21,12 X16,21,17,22 X4,15,1,16 X19,10,20,5
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, {8, 2, -4, -5, 10, -9, 7, 6, -11, -8, 9, -7}

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^4-u v w^3+u v w^2-v w^2+v w-1}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-12} + q^{-11} - q^{-10} +2 q^{-9} + q^{-7} + q^{-6} + q^{-4} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} +2 z^2 a^{12}+4 a^{12} z^{-2} +6 a^{12}-z^6 a^{10}-8 z^4 a^{10}-19 z^2 a^{10}-5 a^{10} z^{-2} -16 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+22 z^2 a^8+2 a^8 z^{-2} +10 a^8 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{15} z^3-2 a^{15} z+a^{15} z^{-1} +a^{14} z^4-2 a^{14} z^2-a^{14} z^{-2} +2 a^{14}-a^{13} z^5+8 a^{13} z^3-13 a^{13} z+5 a^{13} z^{-1} -a^{12} z^6+7 a^{12} z^4-14 a^{12} z^2-4 a^{12} z^{-2} +13 a^{12}+a^{11} z^7-9 a^{11} z^5+26 a^{11} z^3-27 a^{11} z+9 a^{11} z^{-1} +a^{10} z^8-9 a^{10} z^6+27 a^{10} z^4-34 a^{10} z^2-5 a^{10} z^{-2} +20 a^{10}+a^9 z^7-8 a^9 z^5+19 a^9 z^3-16 a^9 z+5 a^9 z^{-1} +a^8 z^8-8 a^8 z^6+21 a^8 z^4-22 a^8 z^2-2 a^8 z^{-2} +10 a^8 }[/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 -7                   1 1 -9                   1 1 -11               1     1 -13           2         2 -15           2 1       1 -17       3 1           2 -19       2 1           1 -21   1 1               0 -23                     0 -25 1                   -1

Integral Khovanov Homology (db, data source)

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