L11n343

L11n342

L11n344

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Link Presentations

[edit Notes on L11n343's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,14,8,15 X13,8,14,9 X15,13,16,22 X17,21,18,20 X21,17,22,16 X19,5,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, -6, 8, -7, 3, -9, 7, -8, 6}

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{(v-1) (w-1) \left(u v-2 u+2 v^2-v\right)}{\sqrt{u} v^{3/2} \sqrt{w}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^3+5 q^2-6 q+8-8 q^{-1} +8 q^{-2} -5 q^{-3} +4 q^{-4} - q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^{-2} +a^6-2 z^2 a^4-2 a^4 z^{-2} -3 a^4+z^4 a^2+a^2 z^{-2} +2 z^4+4 z^2+3-2 z^2 a^{-2} - a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8+a^5 z^7-2 a^3 z^7+3 z^7 a^{-1} +a^6 z^6-12 a^2 z^6+z^6 a^{-2} -10 z^6-2 a^5 z^5+4 a^3 z^5-6 z^5 a^{-1} -5 a^6 z^4-9 a^4 z^4+15 a^2 z^4+4 z^4 a^{-2} +23 z^4-3 a^5 z^3-10 a^3 z^3-a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} +8 a^6 z^2+13 a^4 z^2-8 a^2 z^2-7 z^2 a^{-2} -20 z^2+6 a^5 z+8 a^3 z-4 z a^{-1} -2 z a^{-3} -5 a^6-8 a^4-a^2+2 a^{-2} +5-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                   2 -2 5                 3   3 3               3 2   -1 1             5 3     2 -1           5 5       0 -3         3 3         0 -5       2 5           3 -7     2 3             -1 -9   1 4               3 -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}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/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}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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