L11n196

L11n195

L11n197

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Link Presentations

[edit Notes on L11n196's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X13,21,14,20 X16,9,17,10 X10,19,11,20 X15,7,16,22 X21,15,22,14 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, -3, 7, -6, -4, 11, -8, 5, 3, -7, 6}

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^2 v^3-3 u^2 v^2+4 u^2 v-u^2+u v^4-4 u v^3+7 u v^2-4 u v+u-v^4+4 v^3-3 v^2+v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{2}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} +a^5 z^3-a^5 z-a^5 z^{-1} -a^3 z^5-a^3 z^3-a^3 z-a z^5-a z^3+z^3 a^{-1} -a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^5 z^9-a^3 z^9-a^6 z^8-5 a^4 z^8-4 a^2 z^8-a^5 z^7-8 a^3 z^7-7 a z^7+5 a^4 z^6-2 a^2 z^6-7 z^6-3 a^7 z^5+2 a^5 z^5+16 a^3 z^5+7 a z^5-4 z^5 a^{-1} +2 a^6 z^4+5 a^4 z^4+12 a^2 z^4-z^4 a^{-2} +8 z^4+8 a^7 z^3+4 a^5 z^3-7 a^3 z^3+3 z^3 a^{-1} +a^6 z^2-4 a^4 z^2-7 a^2 z^2-2 z^2-6 a^7 z-4 a^5 z+a^3 z-a z-a^6+a^7 z^{-1} +a^5 z^{-1} }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 3   3 2               4 1   -3 0             6 3     3 -2           7 5       -2 -4         5 5         0 -6       5 7           2 -8     3 5             -2 -10   1 5               4 -12 1 3                 -2 -14 2                   2

Integral Khovanov Homology (db, data source)

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