L11a103

L11a102

L11a104

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Link Presentations

[edit Notes on L11a103's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,6,15,5 X12,4,13,3 X22,14,5,13 X18,9,19,10 X16,11,17,12 X2,16,3,15 X10,17,11,18 X8,19,9,20
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -11, 7, -10, 8, -5, 6, -4, 9, -8, 10, -7, 11, -2, 3, -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{3 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{9/2}+3 q^{7/2}-6 q^{5/2}+7 q^{3/2}-10 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{1}{q^{13/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^5-z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3-2 a^3 z^{-1} +z^5 a+z^3 a+a z^{-1} +z^5 a^{-1} +2 z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} -z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^4 z^{10}-2 a^2 z^{10}-3 a^5 z^9-8 a^3 z^9-5 a z^9-a^6 z^8+7 a^4 z^8+a^2 z^8-7 z^8+17 a^5 z^7+38 a^3 z^7+14 a z^7-7 z^7 a^{-1} +5 a^6 z^6+20 a^2 z^6-7 z^6 a^{-2} +18 z^6-30 a^5 z^5-51 a^3 z^5-5 a z^5+10 z^5 a^{-1} -6 z^5 a^{-3} -7 a^6 z^4-13 a^4 z^4-29 a^2 z^4+8 z^4 a^{-2} -3 z^4 a^{-4} -12 z^4+17 a^5 z^3+15 a^3 z^3-6 a z^3+3 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2+6 a^4 z^2+10 a^2 z^2+7 z^2+a^5 z+7 a^3 z+6 a z-3 z a^{-1} -3 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -2 a^3 z^{-1} -a z^{-1} + a^{-1} z^{-1} + a^{-3} 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                       1 1 8                     2   -2 6                   4 1   3 4                 3 2     -1 2               7 4       3 0             6 5         -1 -2           4 5           -1 -4         5 6             1 -6       2 4               -2 -8     2 5                 3 -10   1 2                   -1 -12   2                     2 -14 1                       -1

Integral Khovanov Homology (db, data source)

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