L11n322

L11n321

L11n323

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Link Presentations

[edit Notes on L11n322's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code	{1, -4, 3, -7}, {-2, -1, 5, -3, -6, 9, -11, 10}, {-9, 2, 4, -5, 7, 6, -8, 11, -10, 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{ -2 q^3+5 q^2-7 q+11-10 q^{-1} +11 q^{-2} -8 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^2 z^6+a^4 z^4-4 a^2 z^4+3 z^4+2 a^4 z^2-8 a^2 z^2-2 z^2 a^{-2} +8 z^2+2 a^4-7 a^2-2 a^{-2} +7+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-9 a^5 z^5+7 a^5 z^3-a^5 z+3 a^4 z^8-5 a^4 z^6-4 a^4 z^4+5 a^4 z^2-a^4 z^{-2} +a^4+a^3 z^9+6 a^3 z^7-22 a^3 z^5+17 a^3 z^3+3 z^3 a^{-3} -6 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +6 a^2 z^8-11 a^2 z^6+z^6 a^{-2} +5 a^2 z^4+4 z^4 a^{-2} -8 a^2 z^2-5 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+3 a^{-2} +a z^9+6 a z^7+3 z^7 a^{-1} -17 a z^5-4 z^5 a^{-1} +17 a z^3+10 z^3 a^{-1} -10 a z-7 z a^{-1} +2 a z^{-1} +3 z^8-4 z^6+10 z^4-15 z^2- z^{-2} +9 }[/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 7                   2 -2 5                 3   3 3               4 2   -2 1             7 3     4 -1           5 6       1 -3         6 5         1 -5       4 7           3 -7     2 4             -2 -9   1 4               3 -11   2                 -2 -13 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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