L11n420

L11n419

L11n421

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Link Presentations

[edit Notes on L11n420's Link Presentations]

Planar diagram presentation	X8192 X7,16,8,17 X3,10,4,11 X17,2,18,3 X18,9,19,10 X11,20,12,21 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 X21,12,22,7
Gauss code	{1, 4, -3, -10, 7, -9}, {-2, -1, 5, 3, -6, 11}, {9, -7, 8, 2, -4, -5, 10, 6, -11, -8}

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^2 v^2 w^3-u^2 v^2 w^2-u v^2 w^3+u v w^2-u v w+u+w-1}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-7} +2 q^{-5} - q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} \left(-z^2\right)-a^{10}+a^8 z^6+6 a^8 z^4+9 a^8 z^2+a^8 z^{-2} +4 a^8-a^6 z^8-7 a^6 z^6-16 a^6 z^4-16 a^6 z^2-2 a^6 z^{-2} -8 a^6+a^4 z^6+6 a^4 z^4+10 a^4 z^2+a^4 z^{-2} +5 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} (-z)-2 a^{10} z^2+a^{10}+a^9 z^7-6 a^9 z^5+8 a^9 z^3-3 a^9 z+2 a^8 z^8-13 a^8 z^6+25 a^8 z^4-20 a^8 z^2-a^8 z^{-2} +8 a^8+a^7 z^9-5 a^7 z^7+3 a^7 z^5+8 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +3 a^6 z^8-20 a^6 z^6+41 a^6 z^4-33 a^6 z^2-2 a^6 z^{-2} +12 a^6+a^5 z^9-6 a^5 z^7+9 a^5 z^5-6 a^5 z+2 a^5 z^{-1} +a^4 z^8-7 a^4 z^6+16 a^4 z^4-15 a^4 z^2-a^4 z^{-2} +6 a^4 }[/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 χ -1                 1 1 -3                   0 -5             2 1   1 -7         1 1 1     1 -9         2 1       1 -11     2 1 1         2 -13     2 1           1 -15 1 1 1             1 -17 1 1               0

Integral Khovanov Homology (db, data source)

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