L11a451

L11a450

L11a452

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Link Presentations

[edit Notes on L11a451's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,10,21,9 X16,8,17,7 X18,12,19,11 X22,16,13,15 X12,18,5,17 X10,20,11,19 X8,22,9,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8, 5, -7}, {11, -2, 6, -4, 7, -5, 8, -3, 9, -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(1) t(2)^3+t(1) t(3) t(2)^3-2 t(3) t(2)^3+t(2)^3+t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+2 t(1) t(2)^2-3 t(1) t(3) t(2)^2+3 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-2 t(3)^3 t(2)-3 t(1) t(3)^2 t(2)+3 t(3)^2 t(2)+2 t(1) t(3) t(2)-t(3) t(2)-t(1) t(3)^3+t(3)^3+2 t(1) t(3)^2-t(3)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^7+3 q^6-4 q^5+7 q^4+ q^{-4} -8 q^3-2 q^{-3} +11 q^2+5 q^{-2} -10 q-7 q^{-1} +9 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4-2 z^2 a^2+z^4-z^2+ z^{-2} +z^4 a^{-2} -2 z^2 a^{-2} -2 a^{-2} z^{-2} -3 a^{-2} +z^4 a^{-4} +z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -z^2 a^{-6} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} +2 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8 a^{-6} +3 z^8+3 a z^7-z^7 a^{-1} -17 z^7 a^{-3} -12 z^7 a^{-5} +z^7 a^{-7} +3 a^2 z^6-16 z^6 a^{-2} -22 z^6 a^{-4} -14 z^6 a^{-6} -5 z^6+2 a^3 z^5-2 a z^5-7 z^5 a^{-1} +12 z^5 a^{-3} +11 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-3 a^2 z^4+29 z^4 a^{-2} +35 z^4 a^{-4} +18 z^4 a^{-6} +8 z^4-2 a^3 z^3+14 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +3 z^3 a^{-7} -2 a^4 z^2+a^2 z^2-28 z^2 a^{-2} -23 z^2 a^{-4} -7 z^2 a^{-6} -9 z^2-11 z a^{-1} -11 z a^{-3} +a^4+13 a^{-2} +8 a^{-4} +5+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                     2   2 11                   2 1   -1 9                 5 2     3 7               5 4       -1 5             6 3         3 3           4 5           1 1         5 6             -1 -1       3 5               2 -3     2 4                 -2 -5     3                   3 -7 1 2                     -1 -9 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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