L10a128

L10a127

L10a129

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Link Presentations

[edit Notes on L10a128's Link Presentations]

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