L10a9

L10a8

L10a10

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Link Presentations

[edit Notes on L10a9's Link Presentations]

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

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)-1) (t(2)-1)^3 \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 9 q^{9/2}-10 q^{7/2}+10 q^{5/2}-\frac{1}{q^{5/2}}-10 q^{3/2}+\frac{2}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+7 \sqrt{q}-\frac{5}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-5} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +10 z a^{-3} +3 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-10 z a^{-1} +2 a z^{-1} -4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-9} +3 z^4 a^{-8} +6 z^5 a^{-7} -4 z^3 a^{-7} +z a^{-7} +9 z^6 a^{-6} -14 z^4 a^{-6} +7 z^2 a^{-6} - a^{-6} +9 z^7 a^{-5} -18 z^5 a^{-5} +10 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -4 z^6 a^{-4} -15 z^4 a^{-4} +14 z^2 a^{-4} -3 a^{-4} +z^9 a^{-3} +11 z^7 a^{-3} -44 z^5 a^{-3} +45 z^3 a^{-3} -19 z a^{-3} +3 a^{-3} z^{-1} +7 z^8 a^{-2} -21 z^6 a^{-2} +11 z^4 a^{-2} +6 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+3 z^7 a^{-1} -5 a z^5-25 z^5 a^{-1} +9 a z^3+39 z^3 a^{-1} -7 a z-21 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^8-8 z^6+9 z^4-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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 4 1   3 10               5 2     -3 8             5 4       1 6           5 5         0 4         5 5           0 2       4 7             3 0     1 3               -2 -2   1 4                 3 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/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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