L11n54

L11n53

L11n55

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Link Presentations

[edit Notes on L11n54's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X7,15,8,14 X15,9,16,8 X13,21,14,20 X17,5,18,22 X21,19,22,18 X19,13,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 6, -8, 7}

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{2 t(2)^5-2 t(2)^4+t(1) t(2)^3+t(2)^2-2 t(1) t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-q^{7/2}+2 q^{5/2}-\frac{1}{q^{5/2}}-3 q^{3/2}+\frac{1}{q^{3/2}}+q^{15/2}-q^{13/2}+2 \sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} -z^5 a^{-3} +a z^3-4 z^3 a^{-1} -4 z^3 a^{-3} +3 a z-4 z a^{-1} -3 z a^{-3} +z a^{-5} +z a^{-7} +2 a z^{-1} -2 a^{-1} z^{-1} - a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -3 z^8 a^{-2} -2 z^8 a^{-4} -z^8-a z^7+4 z^7 a^{-1} +5 z^7 a^{-3} -z^7 a^{-5} -z^7 a^{-7} +16 z^6 a^{-2} +13 z^6 a^{-4} -z^6 a^{-8} +4 z^6+6 a z^5-z^5 a^{-1} -6 z^5 a^{-3} +7 z^5 a^{-5} +6 z^5 a^{-7} -22 z^4 a^{-2} -24 z^4 a^{-4} +2 z^4 a^{-6} +5 z^4 a^{-8} -z^4-11 a z^3-5 z^3 a^{-1} +5 z^3 a^{-3} -9 z^3 a^{-5} -8 z^3 a^{-7} +9 z^2 a^{-2} +17 z^2 a^{-4} -2 z^2 a^{-6} -5 z^2 a^{-8} -5 z^2+8 a z+5 z a^{-1} -4 z a^{-3} +z a^{-5} +2 z a^{-7} -3 a^{-4} + a^{-8} +3-2 a z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} + 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                         0 12                 1 1 1   1 10               2 1       -1 8             1 2 1       0 6           3 3 1         -1 4         1 1 1           1 2       2 4 1             1 0     1 1 1               1 -2     2                   2 -4 1 1                     0 -6 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{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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