L10a151

L10a150

L10a152

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Link Presentations

[edit Notes on L10a151's Link Presentations]

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

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(3)^2-3 t(3)+1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5- q^{-5} +3 q^4+3 q^{-4} -6 q^3-6 q^{-3} +11 q^2+11 q^{-2} -12 q-12 q^{-1} +14 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6+2 a^2 z^4+2 z^4 a^{-2} -3 z^4-a^4 z^2+4 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} -6 z^2-a^4+4 a^2+4 a^{-2} - a^{-4} -6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a z^9+z^9 a^{-1} +3 a^2 z^8+3 z^8 a^{-2} +6 z^8+4 a^3 z^7+9 a z^7+9 z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6+4 a^2 z^6+4 z^6 a^{-2} +3 z^6 a^{-4} +2 z^6+a^5 z^5-4 a^3 z^5-15 a z^5-15 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4-18 a^2 z^4-18 z^4 a^{-2} -6 z^4 a^{-4} -24 z^4-2 a^5 z^3-2 a^3 z^3+2 a z^3+2 z^3 a^{-1} -2 z^3 a^{-3} -2 z^3 a^{-5} +5 a^4 z^2+18 a^2 z^2+18 z^2 a^{-2} +5 z^2 a^{-4} +26 z^2+a^5 z+3 a^3 z+4 a z+4 z a^{-1} +3 z a^{-3} +z a^{-5} -2 a^4-8 a^2-8 a^{-2} -2 a^{-4} -11-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} }[/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 11                     1 -1 9                   2   2 7                 4 1   -3 5               7 2     5 3             5 4       -1 1           9 7         2 -1         7 9           2 -3       4 5             -1 -5     2 7               5 -7   1 4                 -3 -9   2                   2 -11 1                     -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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