L10a165

L10a164

L10a166

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Link Presentations

[edit Notes on L10a165's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X20,16,13,15 X16,20,17,19 X12,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10}, {7, -6, 8, -9, 10, -3, 9, -8}

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 w x-2 u v w-2 u v x+2 u v-2 u w x+2 u w+3 u x-2 u-2 v w x+3 v w+2 v x-2 v+2 w x-2 w-2 x+1}{\sqrt{u} \sqrt{v} \sqrt{w} \sqrt{x}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z^{-3} -4 a^9 z^{-1} -3 a^9 z^{-3} +6 z a^7+8 a^7 z^{-1} +3 a^7 z^{-3} -4 z^3 a^5-6 z a^5-4 a^5 z^{-1} -a^5 z^{-3} +z^5 a^3+z^3 a^3-z^3 a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+4 z^3 a^{11}-6 z a^{11}+4 a^{11} z^{-1} -a^{11} z^{-3} -z^6 a^{10}+6 z^2 a^{10}+3 a^{10} z^{-2} -8 a^{10}-z^7 a^9-3 z^5 a^9+12 z^3 a^9-14 z a^9+9 a^9 z^{-1} -3 a^9 z^{-3} -z^8 a^8-2 z^6 a^8+12 z^2 a^8+6 a^8 z^{-2} -15 a^8-z^9 a^7-6 z^5 a^7+13 z^3 a^7-14 z a^7+9 a^7 z^{-1} -3 a^7 z^{-3} -5 z^8 a^6+9 z^6 a^6-9 z^4 a^6+6 z^2 a^6+3 a^6 z^{-2} -8 a^6-z^9 a^5-5 z^7 a^5+10 z^5 a^5-z^3 a^5-6 z a^5+4 a^5 z^{-1} -a^5 z^{-3} -4 z^8 a^4+6 z^6 a^4-2 z^4 a^4-6 z^7 a^3+13 z^5 a^3-5 z^3 a^3-4 z^6 a^2+7 z^4 a^2-z^5 a+z^3 a }[/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 2                     1 1 0                   3   -3 -2                 4 1   3 -4               5 4     -1 -6             6 3       3 -8           4 5         1 -10         7 6           1 -12       4 10             6 -14     1 1               0 -16     4                 4 -18 1 1                   0 -20 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/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} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=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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