L11n450

L11n449

L11n451

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Link Presentations

[edit Notes on L11n450's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X11,20,12,21 X13,19,14,22 X21,18,22,9 X17,13,18,12 X15,8,16,5 X7,14,8,15 X19,17,20,16 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -8, 7}, {-9, 3, -5, 4}, {-11, 2, -3, 6, -4, 8, -7, 9, -6, 5}

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{(w-1) (x-1)^2 (u v x+u (-v)+u-v x+x-1)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+7 q^{5/2}-13 q^{3/2}+14 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{4}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-a^3 z^5+4 a z^5-2 z^5 a^{-1} -3 a^3 z^3+8 a z^3-6 z^3 a^{-1} +z^3 a^{-3} +a^5 z-7 a^3 z+12 a z-8 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -8 a^3 z^{-1} +11 a z^{-1} -6 a^{-1} z^{-1} + a^{-3} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 10 a^5 z^3-a^5 z^{-3} -11 a^5 z+5 a^5 z^{-1} +6 a^4 z^6+z^6 a^{-4} -4 a^4 z^4-3 z^4 a^{-4} +9 a^4 z^2+3 z^2 a^{-4} +3 a^4 z^{-2} -9 a^4- a^{-4} +13 a^3 z^7+3 z^7 a^{-3} -31 a^3 z^5-8 z^5 a^{-3} +44 a^3 z^3+9 z^3 a^{-3} -3 a^3 z^{-3} -34 a^3 z-6 z a^{-3} +14 a^3 z^{-1} +2 a^{-3} z^{-1} +9 a^2 z^8+4 z^8 a^{-2} -9 a^2 z^6-5 z^6 a^{-2} -9 a^2 z^4-7 z^4 a^{-2} +24 a^2 z^2+14 z^2 a^{-2} +6 a^2 z^{-2} -21 a^2-6 a^{-2} +2 a z^9+2 z^9 a^{-1} +20 a z^7+10 z^7 a^{-1} -62 a z^5-39 z^5 a^{-1} +67 a z^3+42 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -44 a z-27 z a^{-1} +18 a z^{-1} +11 a^{-1} z^{-1} +13 z^8-21 z^6-9 z^4+26 z^2+3 z^{-2} -18 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               5 1   -4 4             8 2     6 2           6 5       -1 0         12 8         4 -2       8 12           4 -4     6 6             0 -6   2 8               6 -8 2 6                 -4 -10 4                   4

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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