L11a324

L11a323

L11a325

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Link Presentations

[edit Notes on L11a324's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,10,17,9 X6,13,7,14 X14,7,15,8 X8,15,1,16 X22,18,9,17 X4,20,5,19 X20,6,21,5 X18,22,19,21
Gauss code	{1, -2, 3, -9, 10, -5, 6, -7}, {4, -1, 2, -3, 5, -6, 7, -4, 8, -11, 9, -10, 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{2 u^3 v^3-2 u^3 v^2-2 u^2 v^3+6 u^2 v^2-3 u^2 v-3 u v^2+6 u v-2 u-2 v+2}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{7/2}-2 q^{5/2}+4 q^{3/2}-7 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{1}{q^{15/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^7-a z^7+a^5 z^5-5 a^3 z^5-5 a z^5+z^5 a^{-1} +4 a^5 z^3-8 a^3 z^3-8 a z^3+4 z^3 a^{-1} +4 a^5 z-5 a^3 z-5 a z+4 z a^{-1} +a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^3-a^9 z+2 a^8 z^4-a^8 z^2+3 a^7 z^5-2 a^7 z^3+a^7 z+4 a^6 z^6-5 a^6 z^4+2 a^6 z^2+5 a^5 z^7-11 a^5 z^5+8 a^5 z^3-5 a^5 z+a^5 z^{-1} +5 a^4 z^8-14 a^4 z^6+10 a^4 z^4-3 a^4 z^2-a^4+3 a^3 z^9-5 a^3 z^7-11 a^3 z^5+18 a^3 z^3-8 a^3 z+a^3 z^{-1} +a^2 z^{10}+3 a^2 z^8+z^8 a^{-2} -24 a^2 z^6-6 z^6 a^{-2} +29 a^2 z^4+12 z^4 a^{-2} -9 a^2 z^2-8 z^2 a^{-2} +5 a z^9+2 z^9 a^{-1} -21 a z^7-11 z^7 a^{-1} +23 a z^5+20 z^5 a^{-1} -8 a z^3-15 z^3 a^{-1} +4 a z+5 z a^{-1} +z^{10}-z^8-12 z^6+24 z^4-11 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                     1   1 4                   3 1   -2 2                 4 1     3 0               3 3       0 -2             6 4         2 -4           4 4           0 -6         4 5             -1 -8       2 4               2 -10     2 4                 -2 -12   1 3                   2 -14   1                     -1 -16 1                       1

Integral Khovanov Homology (db, data source)

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