L11n238

L11n237

L11n239

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Link Presentations

[edit Notes on L11n238's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,16,8,17 X5,1,6,10 X3746 X9,5,10,4 X13,18,14,19 X19,22,20,11 X15,21,16,20 X21,15,22,14 X2,11,3,12 X17,8,18,9
Gauss code	{1, -10, -4, 5, -3, 4, -2, 11, -5, 3}, {10, -1, -6, 9, -8, 2, -11, 6, -7, 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^3 v^3-2 u^3 v^2-2 u^2 v^3+6 u^2 v^2-6 u^2 v+2 u^2+2 u v^3-6 u v^2+6 u v-2 u-2 v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-a^3 z^5+5 a z^5-2 z^5 a^{-1} -4 a^3 z^3+11 a z^3-7 z^3 a^{-1} +z^3 a^{-3} +a^5 z-8 a^3 z+12 a z-9 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -5 a^3 z^{-1} +6 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -5 a^2 z^8-3 z^8 a^{-2} -8 z^8-7 a^3 z^7-14 a z^7-10 z^7 a^{-1} -3 z^7 a^{-3} -3 a^4 z^6+5 a^2 z^6+3 z^6 a^{-2} -z^6 a^{-4} +12 z^6+17 a^3 z^5+44 a z^5+36 z^5 a^{-1} +9 z^5 a^{-3} +9 z^4 a^{-2} +3 z^4 a^{-4} +6 z^4-6 a^5 z^3-29 a^3 z^3-51 a z^3-37 z^3 a^{-1} -9 z^3 a^{-3} -a^4 z^2-3 a^2 z^2-11 z^2 a^{-2} -3 z^2 a^{-4} -10 z^2+7 a^5 z+21 a^3 z+29 a z+19 z a^{-1} +4 z a^{-3} +a^4+a^2+3 a^{-2} + a^{-4} +3-2 a^5 z^{-1} -5 a^3 z^{-1} -6 a z^{-1} -4 a^{-1} z^{-1} - a^{-3} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               4 1   -3 4             6 2     4 2           6 4       -2 0         7 6         1 -2       6 7           1 -4     4 6             -2 -6   2 6               4 -8 1 4                 -3 -10 3                   3

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}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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