L11n283

L11n282

L11n284

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Link Presentations

[edit Notes on L11n283's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 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)^2 (w-1)^2}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+6 q^4-9 q^3- q^{-3} +11 q^2+5 q^{-2} -10 q-6 q^{-1} +11 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-4} +z^2 a^{-4} - a^{-4} z^{-2} - a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -a^2 z^2-2 z^2 a^{-2} +2 a^2 z^{-2} +4 a^{-2} z^{-2} +a^2+4 a^{-2} +2 z^4+2 z^2-5 z^{-2} -4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +5 z^8 a^{-4} +4 z^8+2 a z^7+2 z^7 a^{-3} +4 z^7 a^{-5} -28 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -12 z^6-a z^5-8 z^5 a^{-1} -19 z^5 a^{-3} -12 z^5 a^{-5} +5 a^2 z^4+29 z^4 a^{-2} +11 z^4 a^{-4} -2 z^4 a^{-6} +21 z^4+a^3 z^3+a z^3+6 z^3 a^{-1} +13 z^3 a^{-3} +7 z^3 a^{-5} -3 a^2 z^2-12 z^2 a^{-2} -4 z^2 a^{-4} -11 z^2+5 a z+9 z a^{-1} +5 z a^{-3} +z a^{-5} -3 a^2-2 a^{-2} -4-5 a z^{-1} -9 a^{-1} z^{-1} -5 a^{-3} z^{-1} - a^{-5} z^{-1} +2 a^2 z^{-2} +4 a^{-2} z^{-2} + a^{-4} z^{-2} +5 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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 13                   1 1 11                 3   -3 9               3 1   2 7             6 3     -3 5           5 3       2 3         5 6         1 1       6 5           1 -1     2 7             5 -3   3 4               -1 -5   4                 4 -7 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=1 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/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}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/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}\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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