L11n135

L11n134

L11n136

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Link Presentations

[edit Notes on L11n135's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21
Gauss code	{1, 4, -3, -10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, -4, 2, 10, -8, 11, -9}

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)^2 t(2)^4-t(1)^2 t(2)^3+t(1) t(2)^3-t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+t(1) t(2)-t(2)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+3 q^{7/2}-4 q^{5/2}+3 q^{3/2}-\frac{1}{q^{3/2}}+q^{15/2}-q^{13/2}+2 q^{11/2}-3 \sqrt{q}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-5} +5 z^3 a^{-5} +7 z a^{-5} +2 a^{-5} z^{-1} -z^7 a^{-3} -7 z^5 a^{-3} -17 z^3 a^{-3} -16 z a^{-3} -5 a^{-3} z^{-1} +z^5 a^{-1} +5 z^3 a^{-1} +8 z a^{-1} +3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-3} -z^9 a^{-5} -z^8 a^{-2} -2 z^8 a^{-4} -z^8 a^{-6} +7 z^7 a^{-3} +6 z^7 a^{-5} -z^7 a^{-7} +6 z^6 a^{-2} +11 z^6 a^{-4} +4 z^6 a^{-6} -z^6 a^{-8} -2 z^5 a^{-1} -21 z^5 a^{-3} -15 z^5 a^{-5} +4 z^5 a^{-7} -15 z^4 a^{-2} -23 z^4 a^{-4} -4 z^4 a^{-6} +5 z^4 a^{-8} -z^4-a z^3+6 z^3 a^{-1} +29 z^3 a^{-3} +19 z^3 a^{-5} -3 z^3 a^{-7} +13 z^2 a^{-2} +20 z^2 a^{-4} +2 z^2 a^{-6} -6 z^2 a^{-8} +z^2+2 a z-9 z a^{-1} -20 z a^{-3} -9 z a^{-5} -5 a^{-2} -5 a^{-4} + a^{-8} +3 a^{-1} z^{-1} +5 a^{-3} z^{-1} +2 a^{-5} 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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                     0 12               2 1   -1 10             1       1 8           2 2       0 6         2 1         1 4       1 2           1 2     2 2             0 0     2               2 -2 1 1                 0 -4 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=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/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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