L10a7

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Link Presentations

[edit Notes on L10a7's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,12,15,11 X8493 X12,5,13,6 X20,13,5,14 X16,10,17,9 X10,16,11,15 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 8, -9, 4, -6, 7, -4, 9, -8, 10, -2, 3, -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{(t(1)-1) (t(2)-2) (t(2)-1) (2 t(2)-1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{5}{q^{7/2}}+6 q^{5/2}-\frac{9}{q^{5/2}}-9 q^{3/2}+\frac{11}{q^{3/2}}+\frac{1}{q^{11/2}}+11 \sqrt{q}-\frac{12}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 (-z)+2 a^3 z^3+z^3 a^{-3} +2 a^3 z+a^3 z^{-1} - a^{-3} z^{-1} -a z^5-z^5 a^{-1} -a z^3-z^3 a^{-1} -2 a z-2 a z^{-1} +z a^{-1} +2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a z^9-2 z^9 a^{-1} -4 a^2 z^8-5 z^8 a^{-2} -9 z^8-4 a^3 z^7-2 a z^7-2 z^7 a^{-1} -4 z^7 a^{-3} -4 a^4 z^6+4 a^2 z^6+15 z^6 a^{-2} -z^6 a^{-4} +24 z^6-3 a^5 z^5+8 a z^5+17 z^5 a^{-1} +12 z^5 a^{-3} -a^6 z^4+3 a^4 z^4-2 a^2 z^4-11 z^4 a^{-2} +2 z^4 a^{-4} -19 z^4+4 a^5 z^3+7 a^3 z^3-a z^3-11 z^3 a^{-1} -7 z^3 a^{-3} +a^6 z^2+a^2 z^2+2 z^2 a^{-2} +4 z^2-2 a^5 z-5 a^3 z-6 a z-4 z a^{-1} -z a^{-3} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} + a^{-3} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   3   3 6                 3 1   -2 4               6 3     3 2             5 3       -2 0           7 6         1 -2         6 7           1 -4       3 5             -2 -6     2 6               4 -8   1 3                 -2 -10   2                   2 -12 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=0 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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