L11a366

L11a365

L11a367

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Link Presentations

[edit Notes on L11a366's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,3,15,4 X16,5,17,6 X6,11,7,12 X18,8,19,7 X22,18,11,17 X20,10,21,9 X8,20,9,19 X10,22,1,21 X4,13,5,14 X2,15,3,16
Gauss code	{1, -11, 2, -10, 3, -4, 5, -8, 7, -9}, {4, -1, 10, -2, 11, -3, 6, -5, 8, -7, 9, -6}

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^4 v^3-u^4 v^2+u^3 v^4-3 u^3 v^3+4 u^3 v^2-2 u^3 v-u^2 v^4+4 u^2 v^3-5 u^2 v^2+4 u^2 v-u^2-2 u v^3+4 u v^2-3 u v+u-v^2+v}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{2}{q^{9/2}}+6 q^{7/2}-\frac{4}{q^{7/2}}-9 q^{5/2}+\frac{7}{q^{5/2}}+11 q^{3/2}-\frac{10}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-13 \sqrt{q}+\frac{11}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^5+z^5 a^{-3} +4 a^3 z^3+3 z^3 a^{-3} +4 a^3 z+2 z a^{-3} +a^3 z^{-1} -a z^7-z^7 a^{-1} -5 a z^5-4 z^5 a^{-1} -9 a z^3-4 z^3 a^{-1} -7 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +a^5 z^7-5 a^5 z^5+3 z^5 a^{-5} +7 a^5 z^3-3 z^3 a^{-5} -2 a^5 z+2 a^4 z^8-9 a^4 z^6+5 z^6 a^{-4} +12 a^4 z^4-6 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} +2 a^3 z^9-7 a^3 z^7+6 z^7 a^{-3} +7 a^3 z^5-10 z^5 a^{-3} -5 a^3 z^3+9 z^3 a^{-3} +4 a^3 z-3 z a^{-3} -a^3 z^{-1} +a^2 z^{10}+5 z^8 a^{-2} -8 a^2 z^6-8 z^6 a^{-2} +10 a^2 z^4+6 z^4 a^{-2} -6 a^2 z^2-z^2 a^{-2} +a^2+5 a z^9+3 z^9 a^{-1} -17 a z^7-3 z^7 a^{-1} +24 a z^5-z^5 a^{-1} -23 a z^3+z^3 a^{-1} +10 a z+z a^{-1} -a z^{-1} +z^{10}+3 z^8-12 z^6+11 z^4-5 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 12                       1 -1 10                     2   2 8                   4 1   -3 6                 5 2     3 4               6 4       -2 2             7 5         2 0           5 7           2 -2         5 6             -1 -4       3 6               3 -6     1 4                 -3 -8   1 3                   2 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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