L10a160

L10a159

L10a161

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Link Presentations

[edit Notes on L10a160's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X20,12,13,11 X18,10,19,9 X10,14,11,13 X12,20,7,19 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 4, -5, 3, -6}, {5, -2, 9, -7, 10, -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{u^2 v^2 w^2+u^2 v w^3-2 u^2 v w^2+u^2 v w-u^2 w^3+u^2 w^2-u v^2 w^2+u v^2 w-u v w^3+2 u v w^2-2 u v w+u v-u w^2+u w-v^2 w+v^2-v w^2+2 v w-v-w}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{10}-2 q^9+4 q^8-6 q^7+8 q^6-7 q^5+8 q^4-5 q^3+4 q^2-2 q+1 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -6 z^2 a^{-6} +3 z^2 a^{-8} + a^{-2} +3 a^{-4} -6 a^{-6} +2 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-12} -2 z^2 a^{-12} +2 z^5 a^{-11} -3 z^3 a^{-11} +3 z^6 a^{-10} -6 z^4 a^{-10} +5 z^2 a^{-10} - a^{-10} +3 z^7 a^{-9} -6 z^5 a^{-9} +6 z^3 a^{-9} +3 z^8 a^{-8} -10 z^6 a^{-8} +19 z^4 a^{-8} -14 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +z^9 a^{-7} +z^7 a^{-7} -9 z^5 a^{-7} +16 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} +5 z^8 a^{-6} -20 z^6 a^{-6} +35 z^4 a^{-6} -31 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} +z^9 a^{-5} -8 z^5 a^{-5} +12 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +2 z^8 a^{-4} -6 z^6 a^{-4} +5 z^4 a^{-4} -6 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +2 z^7 a^{-3} -7 z^5 a^{-3} +5 z^3 a^{-3} +z^6 a^{-2} -4 z^4 a^{-2} +4 z^2 a^{-2} - a^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   2 1 -1 17                 2     2 15               4 2     -2 13             4 2       2 11           3 4         1 9         5 4           1 7       2 5             3 5     2 3               -1 3   1 3                 2 1   1                   -1 -1 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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