L10a125

L10a124

L10a126

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Link Presentations

[edit Notes on L10a125's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X20,14,9,13 X18,12,19,11 X12,20,13,19 X14,18,15,17 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {10, -2, 6, -7, 5, -8, 4, -3, 8, -6, 7, -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{u v w^3-2 u v w^2+2 u v w-2 u v-2 u w^3+3 u w^2-3 u w+2 u-2 v w^3+3 v w^2-3 v w+2 v+2 w^3-2 w^2+2 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+7 q^2-10 q+11-11 q^{-1} +11 q^{-2} -6 q^{-3} +5 q^{-4} - q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 a^6 z^{-2} +a^6-3 z^2 a^4-5 a^4 z^{-2} -8 a^4+3 z^4 a^2+9 z^2 a^2+4 a^2 z^{-2} +10 a^2-z^6-3 z^4-4 z^2- z^{-2} -3+z^4 a^{-2} +z^2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-5 a^6 z^4+9 a^6 z^2+2 a^6 z^{-2} -7 a^6+a^5 z^7-a^5 z^5-6 a^5 z^3+11 a^5 z-5 a^5 z^{-1} +a^4 z^8+2 a^4 z^6-14 a^4 z^4+z^4 a^{-4} +20 a^4 z^2+5 a^4 z^{-2} -14 a^4+a^3 z^9+3 a^3 z^5+4 z^5 a^{-3} -17 a^3 z^3-3 z^3 a^{-3} +21 a^3 z-9 a^3 z^{-1} +5 a^2 z^8-7 a^2 z^6+7 z^6 a^{-2} -3 a^2 z^4-8 z^4 a^{-2} +11 a^2 z^2+z^2 a^{-2} +4 a^2 z^{-2} -10 a^2+a z^9+6 a z^7+7 z^7 a^{-1} -8 a z^5-8 z^5 a^{-1} -8 a z^3+13 a z+3 z a^{-1} -5 a z^{-1} - a^{-1} z^{-1} +4 z^8-z^6-3 z^4+z^2+ 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             5 4       1 -1           7 7         0 -3         4 4           0 -5       2 7             5 -7     3 4               -1 -9   1 5                 4 -11                       0 -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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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