L11a165

L11a164

L11a166

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Link Presentations

[edit Notes on L11a165's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X20,18,21,17 X18,12,19,11 X12,20,13,19 X22,16,7,15 X16,22,17,21 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -7, 11, -4, 8, -9, 5, -6, 7, -5, 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{2 u^2 v^4-4 u^2 v^3+3 u^2 v^2+3 u v^3-5 u v^2+3 u v+3 v^2-4 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{6}{q^{9/2}}+q^{7/2}-\frac{8}{q^{7/2}}-2 q^{5/2}+\frac{8}{q^{5/2}}+4 q^{3/2}-\frac{9}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{4}{q^{11/2}}-6 \sqrt{q}+\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^5+4 a^5 z^3+4 a^5 z+a^5 z^{-1} -a^3 z^7-5 a^3 z^5-8 a^3 z^3-5 a^3 z-a z^7-5 a z^5+z^5 a^{-1} -8 a z^3+4 z^3 a^{-1} -6 a z+4 z a^{-1} -2 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-3 a^3 z^9-5 a z^9-2 z^9 a^{-1} -4 a^4 z^8-2 a^2 z^8-z^8 a^{-2} +z^8-5 a^5 z^7+7 a^3 z^7+23 a z^7+11 z^7 a^{-1} -4 a^6 z^6+8 a^4 z^6+20 a^2 z^6+6 z^6 a^{-2} +14 z^6-3 a^7 z^5+12 a^5 z^5+2 a^3 z^5-32 a z^5-19 z^5 a^{-1} -2 a^8 z^4+5 a^6 z^4+2 a^4 z^4-26 a^2 z^4-12 z^4 a^{-2} -33 z^4-a^9 z^3+2 a^7 z^3-11 a^5 z^3-7 a^3 z^3+19 a z^3+12 z^3 a^{-1} +a^8 z^2-2 a^6 z^2-5 a^4 z^2+12 a^2 z^2+9 z^2 a^{-2} +23 z^2+a^9 z-a^7 z+6 a^5 z+4 a^3 z-8 a z-4 z a^{-1} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 a z^{-1} + a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                     1   1 4                   3 1   -2 2                 3 1     2 0               4 3       -1 -2             5 3         2 -4           4 5           1 -6         4 4             0 -8       2 4               2 -10     2 4                 -2 -12     2                   2 -14 1 2                     -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=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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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