L10n110

L10n109

L10n111

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Link Presentations

[edit Notes on L10n110's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,17,14,20 X19,11,20,16 X7,19,8,18 X15,8,16,9 X9,14,10,15 X17,5,18,10 X2536 X11,1,12,4
Gauss code	{1, -9, -2, 10}, {-8, 5, -4, 3}, {9, -1, -5, 6, -7, 8}, {-10, 2, -3, 7, -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(1) t(3)^2-t(1) t(2) t(3)^2-t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2+t(4) t(3)^2-t(1) t(4)^2 t(3)-t(2) t(4)^2 t(3)+t(4)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+2 t(1) t(4) t(3)-2 t(1) t(2) t(4) t(3)+2 t(2) t(4) t(3)-2 t(4) t(3)+t(2) t(4)^2-t(4)^2+t(1) t(2) t(4)-t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -6 q^{9/2}+6 q^{7/2}-10 q^{5/2}+7 q^{3/2}-\frac{3}{q^{3/2}}-q^{13/2}+3 q^{11/2}-8 \sqrt{q}+\frac{4}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} - a^{-5} z^{-3} -z a^{-5} -2 a^{-5} z^{-1} +z^5 a^{-3} +3 z^3 a^{-3} +3 a^{-3} z^{-3} +7 z a^{-3} +7 a^{-3} z^{-1} -4 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a z-9 z a^{-1} +3 a z^{-1} -8 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^8 a^{-2} -2 z^8 a^{-4} -5 z^7 a^{-1} -9 z^7 a^{-3} -4 z^7 a^{-5} -z^6 a^{-2} -z^6 a^{-4} -3 z^6 a^{-6} -3 z^6+17 z^5 a^{-1} +27 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} +11 z^4 a^{-2} +11 z^4 a^{-4} +6 z^4 a^{-6} +6 z^4-6 a z^3-33 z^3 a^{-1} -36 z^3 a^{-3} -7 z^3 a^{-5} +2 z^3 a^{-7} -23 z^2 a^{-2} -13 z^2 a^{-4} -z^2 a^{-6} -11 z^2+11 a z+27 z a^{-1} +25 z a^{-3} +8 z a^{-5} -z a^{-7} +19 a^{-2} +10 a^{-4} +10-5 a z^{-1} -12 a^{-1} z^{-1} -12 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3} }[/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   \ -2 -1 0 1 2 3 4 5 6 χ 14                 1 1 12               2   -2 10             4 1   3 8           4 4     0 6         6 2       4 4       3 6         3 2     5 4           1 0   1 5             4 -2 2 3               -1 -4 3                 3

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