L11n266

L11n265

L11n267

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Link Presentations

[edit Notes on L11n266's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,11,19,12 X20,15,21,16 X22,17,9,18 X16,21,17,22 X12,19,13,20 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, 6, -8, 7, -5, 9, -6, 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{-u v w^3+u v w^2-u v-u w^4+2 u w^3-2 u w^2+u w-v w^4+2 v w^3-2 v w^2+v w+w^5-w^3+w^2}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-8} +2 q^{-7} -5 q^{-6} +5 q^{-5} -5 q^{-4} +6 q^{-3} +q^2-3 q^{-2} +3 q^{-1} +1 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^6-3 z^2 a^6-2 a^6 z^{-2} -5 a^6+z^6 a^4+6 z^4 a^4+16 z^2 a^4+7 a^4 z^{-2} +18 a^4-z^6 a^2-8 z^4 a^2-19 z^2 a^2-8 a^2 z^{-2} -20 a^2+z^4+5 z^2+3 z^{-2} +7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^9-3 z^3 a^9+3 z a^9-a^9 z^{-1} +2 z^6 a^8-4 z^4 a^8+a^8+2 z^7 a^7-2 z^5 a^7-5 z^3 a^7+3 z a^7-a^7 z^{-1} +z^8 a^6-z^4 a^6-6 z^2 a^6-2 a^6 z^{-2} +7 a^6+3 z^7 a^5-9 z^5 a^5+19 z^3 a^5-21 z a^5+7 a^5 z^{-1} +z^8 a^4-5 z^6 a^4+18 z^4 a^4-27 z^2 a^4-7 a^4 z^{-2} +22 a^4+2 z^7 a^3-16 z^5 a^3+47 z^3 a^3-45 z a^3+15 a^3 z^{-1} +z^8 a^2-11 z^6 a^2+36 z^4 a^2-45 z^2 a^2-8 a^2 z^{-2} +28 a^2+z^7 a-10 z^5 a+26 z^3 a-24 z a+8 a z^{-1} +z^8-8 z^6+21 z^4-24 z^2-3 z^{-2} +13 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                       1 1 3                       1 1 1                   1     1 -1               4         4 -3             3 4 1       0 -5           4 1           3 -7         1 3 1           1 -9       4 4               0 -11     1 1                 0 -13   1 4                   -3 -15   1                     1 -17 1                       -1

Integral Khovanov Homology (db, data source)

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