L10a139

L10a138

L10a140

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Link Presentations

[edit Notes on L10a139's Link Presentations]

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

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(2)-1) \left(t(3) t(2)^2-2 t(2)^2-3 t(1) t(2)+2 t(1) t(3) t(2)-3 t(3) t(2)+2 t(2)+t(1)-2 t(1) t(3)\right)}{\sqrt{t(1)} t(2)^{3/2} \sqrt{t(3)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -3 q^{-5} +q^4+6 q^{-4} -2 q^3-8 q^{-3} +5 q^2+10 q^{-2} -7 q-10 q^{-1} +11 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6-3 z^2 a^4-2 a^4+2 z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2+z^4-2 z^2-2 z^{-2} -3-2 z^2 a^{-2} + a^{-2} z^{-2} + a^{-4} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-9 a^5 z^5+7 a^5 z^3-2 a^5 z+3 a^4 z^8-5 a^4 z^6-5 a^4 z^4+z^4 a^{-4} +6 a^4 z^2-2 z^2 a^{-4} -a^4+ a^{-4} +a^3 z^9+6 a^3 z^7-24 a^3 z^5+2 z^5 a^{-3} +21 a^3 z^3-2 z^3 a^{-3} -6 a^3 z+6 a^2 z^8-11 a^2 z^6+3 z^6 a^{-2} -a^2 z^4-3 z^4 a^{-2} +8 a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-3 a^{-2} +a z^9+6 a z^7+3 z^7 a^{-1} -17 a z^5+11 a z^3-5 z^3 a^{-1} +2 a z+6 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-2 z^6-3 z^4+10 z^2+2 z^{-2} -7 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   2 1 -1 5                 3     3 3               4 2     -2 1             7 3       4 -1           5 6         1 -3         5 5           0 -5       3 5             2 -7     3 5               -2 -9   1 4                 3 -11   2                   -2 -13 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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