L11n221

L11n220

L11n222

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Link Presentations

[edit Notes on L11n221's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X12,4,13,3 X22,16,9,15 X2,17,3,18 X21,4,22,5 X5,15,6,14 X13,21,14,20 X16,12,17,11 X6,19,7,20 X18,7,19,8
Gauss code	{1, -5, 3, 6, -7, -10, 11, -2}, {2, -1, 9, -3, -8, 7, 4, -9, 5, -11, 10, 8, -6, -4}

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{t(2)^2 t(1)^3-t(2) t(1)^3+2 t(2)^3 t(1)^2-6 t(2)^2 t(1)^2+6 t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+6 t(2)^2 t(1)-6 t(2) t(1)+2 t(1)-t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{5}{q^{7/2}}+6 q^{5/2}-\frac{9}{q^{5/2}}-9 q^{3/2}+\frac{10}{q^{3/2}}+11 \sqrt{q}-\frac{12}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 a^3 z^3+z^3 a^{-3} +3 a^3 z+z a^{-3} +2 a^3 z^{-1} -2 a z^5-z^5 a^{-1} -6 a z^3-z^3 a^{-1} -7 a z+z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a z^9-2 z^9 a^{-1} -4 a^2 z^8-4 z^8 a^{-2} -8 z^8-3 a^3 z^7-3 z^7 a^{-3} -a^4 z^6+7 a^2 z^6+11 z^6 a^{-2} -z^6 a^{-4} +20 z^6-a z^5+8 z^5 a^{-1} +9 z^5 a^{-3} -4 a^4 z^4-11 a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} -18 z^4-3 a^5 z^3+3 a^3 z^3+11 a z^3-2 z^3 a^{-1} -7 z^3 a^{-3} +2 a^4 z^2+9 a^2 z^2+3 z^2 a^{-2} -2 z^2 a^{-4} +12 z^2+2 a^5 z-5 a^3 z-10 a z-2 z a^{-1} +z a^{-3} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               4 1   -3 4             5 2     3 2           6 4       -2 0         6 5         1 -2       5 7           2 -4     4 5             -1 -6   1 5               4 -8 1 4                 -3 -10 2                   2

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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