L11n164

L11n163

L11n165

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Link Presentations

[edit Notes on L11n164's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X12,17,13,18 X14,5,15,6 X4,13,5,14 X18,11,19,12 X19,7,20,22 X15,21,16,20 X21,17,22,16 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 6, -3, 5, -4, -8, 9, 3, -6, -7, 8, -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^2 v^3-u^2 v^2+2 u^2 v-u^2+u v^4-2 u v^3+3 u v^2-2 u v+u-v^4+2 v^3-v^2+v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{6}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^3+2 a^7 z+a^7 z^{-1} -a^5 z^5-3 a^5 z^3-3 a^5 z-a^5 z^{-1} -a^3 z^5-3 a^3 z^3-3 a^3 z+a z^3+a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-5 a^9 z^5+8 a^9 z^3-4 a^9 z+2 a^8 z^8-9 a^8 z^6+11 a^8 z^4-3 a^8 z^2+a^7 z^9-a^7 z^7-8 a^7 z^5+10 a^7 z^3-a^7 z^{-1} +4 a^6 z^8-14 a^6 z^6+11 a^6 z^4-3 a^6 z^2+a^6+a^5 z^9-7 a^5 z^5+3 a^5 z^3+2 a^5 z-a^5 z^{-1} +2 a^4 z^8-4 a^4 z^6+2 a^3 z^7-4 a^3 z^5+4 a^3 z^3-3 a^3 z+a^2 z^6+a^2 z^2+3 a z^3-a z+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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   1 -1 0                 2   2 -2               3 2   -1 -4             3 1     2 -6           3 3       0 -8         3 3         0 -10       2 3           1 -12     2 3             -1 -14   1 3               2 -16   1                 -1 -18 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_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}_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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