L10a62

L10a61

L10a63

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Link Presentations

[edit Notes on L10a62's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X16,12,17,11 X14,5,15,6 X4,15,5,16 X18,14,19,13 X12,18,13,17 X6,20,1,19
Gauss code	{1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 5, -9, 8, -6, 7, -5, 9, -8, 10, -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{u^2 v^4-2 u^2 v^3+2 u^2 v^2-2 u^2 v+u^2-2 u v^4+3 u v^3-3 u v^2+3 u v-2 u+v^4-2 v^3+2 v^2-2 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 8 q^{9/2}-9 q^{7/2}+9 q^{5/2}-\frac{1}{q^{5/2}}-9 q^{3/2}+\frac{2}{q^{3/2}}-q^{15/2}+3 q^{13/2}-5 q^{11/2}+6 \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} -2 z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +9 z^3 a^{-3} +7 z a^{-3} + a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-9 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-9} +3 z^4 a^{-8} -z^2 a^{-8} +5 z^5 a^{-7} -3 z^3 a^{-7} +7 z^6 a^{-6} -10 z^4 a^{-6} +4 z^2 a^{-6} +7 z^7 a^{-5} -14 z^5 a^{-5} +9 z^3 a^{-5} -4 z a^{-5} +4 z^8 a^{-4} -3 z^6 a^{-4} -13 z^4 a^{-4} +11 z^2 a^{-4} - a^{-4} +z^9 a^{-3} +8 z^7 a^{-3} -35 z^5 a^{-3} +36 z^3 a^{-3} -13 z a^{-3} + a^{-3} z^{-1} +6 z^8 a^{-2} -18 z^6 a^{-2} +8 z^4 a^{-2} +7 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+2 z^7 a^{-1} -5 a z^5-21 z^5 a^{-1} +9 a z^3+32 z^3 a^{-1} -7 a z-16 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^8-8 z^6+8 z^4+z^2-3 }[/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                 3 1   2 10               5 2     -3 8             4 3       1 6           5 5         0 4         4 4           0 2       3 6             3 0     2 3               -1 -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}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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