L10n61

L10n60

L10n62

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Link Presentations

[edit Notes on L10n61's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,7,17,8 X5,1,6,10 X3746 X9,5,10,4 X20,17,11,18 X18,13,19,14 X14,19,15,20 X2,11,3,12 X8,15,9,16
Gauss code	{1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, {9, -1, 7, -8, 10, -2, 6, -7, 8, -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{-2 u^3 v^2+2 u^3 v-u^3+2 u^2 v^2-2 u^2 v-2 u v^2+2 u v-v^3+2 v^2-2 v}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{6}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{4}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7-2 a^7 z^{-1} +3 z^3 a^5+9 z a^5+7 a^5 z^{-1} -2 z^5 a^3-9 z^3 a^3-14 z a^3-7 a^3 z^{-1} +2 z^3 a+4 z a+2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-4 a^8 z^4+5 a^8 z^2-2 a^8+2 a^7 z^7-7 a^7 z^5+7 a^7 z^3-4 a^7 z+2 a^7 z^{-1} +a^6 z^8+2 a^6 z^6-17 a^6 z^4+20 a^6 z^2-8 a^6+6 a^5 z^7-20 a^5 z^5+22 a^5 z^3-17 a^5 z+7 a^5 z^{-1} +a^4 z^8+5 a^4 z^6-24 a^4 z^4+29 a^4 z^2-13 a^4+4 a^3 z^7-12 a^3 z^5+18 a^3 z^3-16 a^3 z+7 a^3 z^{-1} +4 a^2 z^6-11 a^2 z^4+17 a^2 z^2-8 a^2+a z^5+3 a z^3-3 a z+2 a z^{-1} +3 z^2-2 }[/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 χ 2                 2 -2 0               2   2 -2             4 3   -1 -4           3 1     2 -6         2 4       2 -8       4 3         1 -10     1 3           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=-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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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