L11n290

L11n289

L11n291

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Link Presentations

[edit Notes on L11n290's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X4,15,1,16 X17,22,18,11 X21,16,22,17
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -7, 8}, {-6, 2, 4, -5, 9, 11, -10, 6, -8, 7, -11, 10}

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-1) (v-1) (w-1) \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q-1+6 q^{-1} -6 q^{-2} +9 q^{-3} -8 q^{-4} +7 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^4\right)-2 a^6 z^2-a^6 z^{-2} -2 a^6+a^4 z^6+4 a^4 z^4+7 a^4 z^2+4 a^4 z^{-2} +7 a^4-2 a^2 z^4-6 a^2 z^2-5 a^2 z^{-2} -8 a^2+z^2+2 z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-6 z^4 a^8+z^2 a^8+4 z^7 a^7-8 z^5 a^7+2 z^3 a^7-2 z a^7+a^7 z^{-1} +3 z^8 a^6-6 z^6 a^6+5 z^4 a^6-5 z^2 a^6-a^6 z^{-2} +3 a^6+z^9 a^5+z^7 a^5-6 z^5 a^5+13 z^3 a^5-13 z a^5+5 a^5 z^{-1} +4 z^8 a^4-14 z^6 a^4+26 z^4 a^4-20 z^2 a^4-4 a^4 z^{-2} +10 a^4+z^9 a^3-3 z^7 a^3+4 z^5 a^3+10 z^3 a^3-17 z a^3+9 a^3 z^{-1} +z^8 a^2-5 z^6 a^2+16 z^4 a^2-18 z^2 a^2-5 a^2 z^{-2} +11 a^2+z^5 a+z^3 a-7 z a+5 a z^{-1} +z^4-4 z^2-2 z^{-2} +5 }[/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 χ 3                   1 1 1                     0 -1               6 1   5 -3             4 4     0 -5           5 2       3 -7         3 4         1 -9       4 5           -1 -11     2 3             1 -13   1 4               -3 -15   2                 2 -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=-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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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]

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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