L11n281

L11n280

L11n282

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Link Presentations

[edit Notes on L11n281's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 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
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-1) (v-1) (w-1) \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{-2} -3 q^{-3} +8 q^{-4} -7 q^{-5} +9 q^{-6} -8 q^{-7} +6 q^{-8} -4 q^{-9} + q^{-10} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} +a^8 z^4+2 a^8 z^2+4 a^8 z^{-2} +4 a^8-a^6 z^6-4 a^6 z^4-8 a^6 z^2-5 a^6 z^{-2} -10 a^6+2 a^4 z^4+6 a^4 z^2+2 a^4 z^{-2} +6 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{12}+4 z^5 a^{11}-4 z^3 a^{11}+a^{11} z^{-1} +6 z^6 a^{10}-8 z^4 a^{10}+2 z^2 a^{10}-a^{10} z^{-2} +4 z^7 a^9-z^5 a^9-3 z^3 a^9-5 z a^9+5 a^9 z^{-1} +z^8 a^8+7 z^6 a^8-11 z^4 a^8+z^2 a^8-4 a^8 z^{-2} +5 a^8+5 z^7 a^7-6 z^5 a^7+7 z^3 a^7-15 z a^7+9 a^7 z^{-1} +z^8 a^6+z^6 a^6+z^4 a^6-8 z^2 a^6-5 a^6 z^{-2} +10 a^6+z^7 a^5-z^5 a^5+6 z^3 a^5-10 z a^5+5 a^5 z^{-1} +3 z^4 a^4-7 z^2 a^4-2 a^4 z^{-2} +6 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 2 2 -5               3 2 -1 -7             5     5 -9           2 3     1 -11         7 5       2 -13       3 4         1 -15     3 5           -2 -17   1 3             2 -19   3               -3 -21 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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