L10a44

L10a43

L10a45

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Link Presentations

[edit Notes on L10a44's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,8,17,7 X18,10,19,9 X20,12,5,11 X8,18,9,17 X10,20,11,19 X12,16,13,15 X2536 X4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 4, -7, 5, -8, 10, -2, 8, -3, 6, -4, 7, -5}

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