L10a97

L10a96

L10a98

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Link Presentations

[edit Notes on L10a97's Link Presentations]

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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