L11n408

L11n407

L11n409

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Link Presentations

[edit Notes on L11n408's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,16,8,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {-6, 5, -7, 4}, {10, -1, -3, 9, -8, 6, -5, -2, 11, 8, -9, 3, -4, 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-1) (v-1) (w-1)}{\sqrt{u} \sqrt{v} \sqrt{w}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-q^2+q+2- q^{-1} +3 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^2\right)-a^6+a^4 z^4+2 a^4 z^2+a^4+a^2 z^4+3 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^2+2 a^{-2} -z^4-5 z^2-2 z^{-2} -5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-3 a^5 z^7-5 a^3 z^7+z^7 a^{-1} -10 a^6 z^6-15 a^4 z^6-5 a^2 z^6+z^6 a^{-2} +z^6-5 a^7 z^5-2 a^5 z^5+7 a^3 z^5-a z^5-5 z^5 a^{-1} +14 a^6 z^4+21 a^4 z^4+2 a^2 z^4-5 z^4 a^{-2} -10 z^4+7 a^7 z^3+6 a^5 z^3-7 a^3 z^3-3 a z^3+3 z^3 a^{-1} -7 a^6 z^2-13 a^4 z^2+7 a^2 z^2+6 z^2 a^{-2} +19 z^2-2 a^7 z-3 a^5 z+3 a^3 z+7 a z+3 z a^{-1} +2 a^6+2 a^4-6 a^2-4 a^{-2} -9-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                         0 3                   1 1   0 1               4 1       3 -1             2 4 1       1 -3           2 2 2         2 -5         2 2             0 -7       2 2 1             1 -9     1 2                 1 -11   1 2                   -1 -13   1                     1 -15 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{ 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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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{ {\mathbb Z} }[/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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