L11n325

L11n324

L11n326

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Link Presentations

[edit Notes on L11n325's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,21,12,20 X19,22,20,13 X13,12,14,5 X4,17,1,18 X21,11,22,10
Gauss code	{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11, -7, 9}, {-9, 2, 4, -5, 10, 6, -8, 7, -11, 8}

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^{-8} +3 q^{-7} -6 q^{-6} +9 q^{-5} -10 q^{-4} +12 q^{-3} -9 q^{-2} +2 q+8 q^{-1} -4 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^6-2 z^2 a^6-2 a^6+z^6 a^4+4 z^4 a^4+8 z^2 a^4+a^4 z^{-2} +6 a^4-3 z^4 a^2-8 z^2 a^2-2 a^2 z^{-2} -7 a^2+2 z^2+ z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+4 z^7 a^7-6 z^5 a^7+z a^7+3 z^8 a^6-z^6 a^6-6 z^4 a^6+4 z^2 a^6+z^9 a^5+5 z^7 a^5-11 z^5 a^5+8 z^3 a^5-3 z a^5+5 z^8 a^4-8 z^6 a^4+10 z^4 a^4-11 z^2 a^4-a^4 z^{-2} +7 a^4+z^9 a^3+2 z^7 a^3-3 z^5 a^3+5 z^3 a^3-6 z a^3+2 a^3 z^{-1} +2 z^8 a^2-4 z^6 a^2+13 z^4 a^2-18 z^2 a^2-2 a^2 z^{-2} +9 a^2+z^7 a+z^5 a-z^3 a-3 z a+2 a z^{-1} +3 z^4-6 z^2- z^{-2} +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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   2 2 1                 2   -2 -1               6 2   4 -3             5 4     -1 -5           7 4       3 -7         5 7         2 -9       4 5           -1 -11     2 5             3 -13   1 4               -3 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

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