L10a122

L10a121

L10a123

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Link Presentations

[edit Notes on L10a122's Link Presentations]

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

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)^3+t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+4 t(1) t(3)^2-3 t(1) t(2) t(3)^2+6 t(2) t(3)^2-3 t(3)^2-6 t(1) t(3)+3 t(1) t(2) t(3)-4 t(2) t(3)+3 t(3)+2 t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -2 q^{-5} +q^4+6 q^{-4} -4 q^3-9 q^{-3} +9 q^2+13 q^{-2} -11 q-14 q^{-1} +14 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^6-3 z^2 a^4-3 a^4 z^{-2} -5 a^4+3 z^4 a^2+7 z^2 a^2+4 a^2 z^{-2} +8 a^2-z^6-3 z^4-6 z^2-3 z^{-2} -6+z^4 a^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-4 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -4 a^6+2 a^5 z^7-5 a^5 z^5+3 a^5 z^3+a^5 z-a^5 z^{-1} +2 a^4 z^8+a^4 z^6-16 a^4 z^4+z^4 a^{-4} +24 a^4 z^2+3 a^4 z^{-2} -14 a^4+a^3 z^9+7 a^3 z^7-20 a^3 z^5+4 z^5 a^{-3} +12 a^3 z^3-z^3 a^{-3} +a^3 z-a^3 z^{-1} +7 a^2 z^8-2 a^2 z^6+9 z^6 a^{-2} -30 a^2 z^4-11 z^4 a^{-2} +40 a^2 z^2+6 z^2 a^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} -21 a^2-4 a^{-2} +a z^9+15 a z^7+10 z^7 a^{-1} -32 a z^5-13 z^5 a^{-1} +14 a z^3+4 z^3 a^{-1} +a z+z a^{-1} -a z^{-1} - a^{-1} z^{-1} +5 z^8+7 z^6-30 z^4+28 z^2+3 z^{-2} -14 }[/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                   3   -3 5                 6 1   5 3               5 3     -2 1             9 6       3 -1           8 8         0 -3         5 6           -1 -5       4 8             4 -7     2 5               -3 -9   1 5                 4 -11   1                   -1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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