L11n345

L11n344

L11n346

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Link Presentations

[edit Notes on L11n345's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X11,19,12,18 X16,8,17,7 X17,21,18,20 X19,5,20,12 X8,22,9,21 X22,10,13,9 X10,14,11,13 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, -3, 6}, {9, -2, 11, -4, -5, 3, -6, 5, 7, -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{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3+t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+t(1) t(3) t(2)^2-t(3)^2 t(2)+t(2)+t(1)-t(1) t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{10}-q^8+q^7-q^6+3 q^5-2 q^4+3 q^3-q^2+q }[/math] (db)
Signature	6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -20 z^2 a^{-6} +9 z^2 a^{-8} -z^2 a^{-10} +7 a^{-4} -11 a^{-6} +4 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-12} -4 z^2 a^{-12} + a^{-12} -z^3 a^{-11} -z^4 a^{-10} -z^2 a^{-10} +2 z^7 a^{-9} -11 z^5 a^{-9} +11 z^3 a^{-9} +3 z^8 a^{-8} -18 z^6 a^{-8} +29 z^4 a^{-8} -16 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +z^9 a^{-7} -3 z^7 a^{-7} -7 z^5 a^{-7} +20 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +4 z^8 a^{-6} -25 z^6 a^{-6} +48 z^4 a^{-6} -37 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +z^9 a^{-5} -5 z^7 a^{-5} +4 z^5 a^{-5} +8 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +17 z^4 a^{-4} -18 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4} }[/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 8 χ 21                     1 1 19                     1 1 17               2 1     -1 15             1 1       0 13           2 3 1       0 11         1 1 2         2 9       1 2             1 7     2 1               1 5   1 3                 2 3                       0 1 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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