L10a138

L10a137

L10a139

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Link Presentations

[edit Notes on L10a138's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X20,12,13,11 X12,14,5,13 X4,17,1,18 X10,20,11,19
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 6, -10, 7, -8}, {8, -2, 4, -5, 9, -6, 10, -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)-1) (t(3)-1) \left(t(2)^2 t(3)^2+t(2) t(3)+1\right)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-4 q^7+6 q^6-7 q^5+8 q^4-6 q^3+6 q^2-3 q+3- q^{-1} }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-6} -4 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} z^{-2} + a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +11 z^4 a^{-4} +5 z^2 a^{-4} -2 a^{-4} z^{-2} -4 a^{-4} -z^6 a^{-2} -4 z^4 a^{-2} -2 z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-11} +3 z^4 a^{-10} -2 z^2 a^{-10} +4 z^5 a^{-9} -3 z^3 a^{-9} +4 z^6 a^{-8} -3 z^4 a^{-8} -2 z^2 a^{-8} + a^{-8} +4 z^7 a^{-7} -6 z^5 a^{-7} +4 z^8 a^{-6} -12 z^6 a^{-6} +12 z^4 a^{-6} -6 z^2 a^{-6} + a^{-6} z^{-2} - a^{-6} +2 z^9 a^{-5} -4 z^7 a^{-5} -3 z^5 a^{-5} +3 z^3 a^{-5} +4 z a^{-5} -2 a^{-5} z^{-1} +7 z^8 a^{-4} -31 z^6 a^{-4} +39 z^4 a^{-4} -13 z^2 a^{-4} +2 a^{-4} z^{-2} -4 a^{-4} +2 z^9 a^{-3} -7 z^7 a^{-3} +3 z^5 a^{-3} +2 z^3 a^{-3} +4 z a^{-3} -2 a^{-3} z^{-1} +3 z^8 a^{-2} -15 z^6 a^{-2} +21 z^4 a^{-2} -7 z^2 a^{-2} + a^{-2} z^{-2} -3 a^{-2} +z^7 a^{-1} -4 z^5 a^{-1} +3 z^3 a^{-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   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                     1 -1 17                   2   2 15                 2 1   -1 13               4 2     2 11             5 4       -1 9           3 2         1 7         3 5           2 5       3 3             0 3     2 5               3 1   1 1                 0 -1   2                   2 -3 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{ 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}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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