L10a147

L10a146

L10a148

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Link Presentations

[edit Notes on L10a147's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X20,10,13,9 X18,8,19,7 X8,14,9,13 X12,15,5,16 X10,20,11,19 X2,11,3,12 X4,18,1,17
Gauss code	{1, -9, 2, -10}, {3, -1, 5, -6, 4, -8, 9, -7}, {6, -2, 7, -3, 10, -5, 8, -4}

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(3)^2 t(2)^3-t(3)^2 t(2)^3-t(1) t(3) t(2)^3+t(3) t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-3 t(1) t(3)^2 t(2)^2+3 t(3)^2 t(2)^2-t(1) t(2)^2+4 t(1) t(3) t(2)^2-3 t(3) t(2)^2+t(2)^2-t(1) t(3)^3 t(2)+t(3)^3 t(2)+3 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+t(1) t(2)-3 t(1) t(3) t(2)+3 t(3) t(2)-t(2)-t(1) t(3)^2+t(3)^2+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+6 q^4+ q^{-4} -9 q^3-4 q^{-3} +14 q^2+8 q^{-2} -13 q-11 q^{-1} +14 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -2 z^4+a^2 z^2-4 z^2 a^{-2} +2 z^2 a^{-4} -3 a^{-2} + a^{-4} +2-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +10 z^8 a^{-2} +4 z^8 a^{-4} +6 z^8+9 a z^7+8 z^7 a^{-1} +2 z^7 a^{-3} +3 z^7 a^{-5} +8 a^2 z^6-23 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -3 z^6+4 a^3 z^5-11 a z^5-19 z^5 a^{-1} -13 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-9 a^2 z^4+20 z^4 a^{-2} +11 z^4 a^{-4} -3 z^4 a^{-6} -4 z^4-2 a^3 z^3+3 a z^3+8 z^3 a^{-1} +10 z^3 a^{-3} +7 z^3 a^{-5} +3 a^2 z^2-13 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -z^2-3 z a^{-1} -3 z a^{-3} +5 a^{-2} +3 a^{-4} +3+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 χ 13                     1 1 11                   2   -2 9                 4 1   3 7               6 3     -3 5             8 3       5 3           5 6         1 1         9 8           1 -1       5 8             3 -3     3 6               -3 -5   1 5                 4 -7   3                   -3 -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{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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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