L11a356

L11a355

L11a357

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Link Presentations

[edit Notes on L11a356's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,5,17,6 X14,3,15,4 X20,10,21,9 X18,8,19,7 X22,17,11,18 X8,20,9,19 X4,15,5,16 X6,22,7,21 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 3, -8, 2, -9, 5, -7, 4, -11}, {10, -1, 11, -3, 8, -2, 6, -5, 7, -4, 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^3-7 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{ -\frac{8}{q^{9/2}}-q^{7/2}+\frac{10}{q^{7/2}}+3 q^{5/2}-\frac{13}{q^{5/2}}-6 q^{3/2}+\frac{13}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{4}{q^{11/2}}+9 \sqrt{q}-\frac{12}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+a z^7-a^5 z^5+5 a^3 z^5+4 a z^5-z^5 a^{-1} -4 a^5 z^3+9 a^3 z^3+4 a z^3-3 z^3 a^{-1} -4 a^5 z+6 a^3 z-a z-2 z a^{-1} +a^3 z^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-4 a^8 z^4+4 a^8 z^2+2 a^7 z^7-7 a^7 z^5+7 a^7 z^3-2 a^7 z+2 a^6 z^8-4 a^6 z^6+a^6 z^2+2 a^5 z^9-5 a^5 z^7+9 a^5 z^5-12 a^5 z^3+3 a^5 z+a^4 z^{10}-a^4 z^6+2 a^4 z^4-4 a^4 z^2+5 a^3 z^9-16 a^3 z^7+30 a^3 z^5+z^5 a^{-3} -27 a^3 z^3-2 z^3 a^{-3} +10 a^3 z-a^3 z^{-1} +a^2 z^{10}+3 a^2 z^8-11 a^2 z^6+3 z^6 a^{-2} +16 a^2 z^4-6 z^4 a^{-2} -6 a^2 z^2+z^2 a^{-2} +a^2+3 a z^9-4 a z^7+5 z^7 a^{-1} +a z^5-12 z^5 a^{-1} +3 a z^3+9 z^3 a^{-1} +2 a z-3 z a^{-1} -a z^{-1} +5 z^8-12 z^6+12 z^4-4 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   4 1   3 2                 5 2     -3 0               7 4       3 -2             7 6         -1 -4           6 6           0 -6         5 8             3 -8       3 5               -2 -10     1 5                 4 -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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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