L11a221

L11a220

L11a222

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Link Presentations

[edit Notes on L11a221's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X16,6,17,5 X20,13,21,14 X18,15,19,16 X14,19,15,20 X22,10,7,9 X4,18,5,17 X10,22,11,21 X2738 X6,11,1,12
Gauss code	{1, -10, 2, -8, 3, -11}, {10, -1, 7, -9, 11, -2, 4, -6, 5, -3, 8, -5, 6, -4, 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 v-2 u-2 v+2) (2 u v-2 u-2 v+1)}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{10}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+3 q^{5/2}-\frac{16}{q^{5/2}}-6 q^{3/2}+\frac{15}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+10 \sqrt{q}-\frac{14}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)+2 a^5 z^3+a^5 z-a^3 z^5+a^3 z^{-1} -z a^{-3} -a z^5+2 z^3 a^{-1} -a z-a z^{-1} +z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-6 a^3 z^9-3 a z^9-4 a^6 z^8-6 a^4 z^8-6 a^2 z^8-4 z^8-3 a^7 z^7+3 a^5 z^7+11 a^3 z^7+a z^7-4 z^7 a^{-1} -a^8 z^6+10 a^6 z^6+20 a^4 z^6+15 a^2 z^6-3 z^6 a^{-2} +3 z^6+9 a^7 z^5+6 a^5 z^5-8 a^3 z^5+a z^5+5 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-6 a^6 z^4-20 a^4 z^4-15 a^2 z^4+6 z^4 a^{-2} +2 z^4-7 a^7 z^3-6 a^5 z^3+3 a^3 z^3+2 z^3 a^{-3} -2 a^8 z^2+2 a^6 z^2+7 a^4 z^2+5 a^2 z^2-3 z^2 a^{-2} -z^2+2 a^7 z+a^5 z-3 a^3 z-a z-z a^{-3} -a^2+a^3 z^{-1} +a 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   4 1   3 2                 6 2     -4 0               8 4       4 -2             8 7         -1 -4           8 7           1 -6         6 9             3 -8       4 7               -3 -10     2 6                 4 -12   1 4                   -3 -14   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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