L11n437

L11n436

L11n438

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Link Presentations

[edit Notes on L11n437's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X13,20,14,21 X9,18,10,19 X21,10,22,11 X19,14,20,7 X5,17,6,16 X17,22,18,15 X2738 X4,11,5,12 X15,1,16,6
Gauss code	{1, -9, 2, -10, -7, 11}, {9, -1, -4, 5, 10, -2, -3, 6}, {-11, 7, -8, 4, -6, 3, -5, 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 w+v) \left(u v^2 w-u v^2+u-v^2 w+w-1\right)}{u v^{3/2} w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1- q^{-1} + q^{-2} + q^{-4} + q^{-5} + q^{-7} - q^{-8} + q^{-9} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^2+a^8 z^{-2} +a^8-2 a^6 z^{-2} -3 a^6-a^4 z^6-5 a^4 z^4-5 a^4 z^2+a^4 z^{-2} +a^2 z^4+4 a^2 z^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-5 a^{10} z^4+5 a^{10} z^2+a^9 z^7-5 a^9 z^5+5 a^9 z^3+a^8 z^8-7 a^8 z^6+16 a^8 z^4-17 a^8 z^2-a^8 z^{-2} +7 a^8+a^7 z^7-7 a^7 z^5+13 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +a^6 z^8-8 a^6 z^6+21 a^6 z^4-25 a^6 z^2-2 a^6 z^{-2} +11 a^6+a^5 z^7-7 a^5 z^5+13 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +a^4 z^6-5 a^4 z^4+3 a^4 z^2-a^4 z^{-2} +3 a^4+a^3 z^7-5 a^3 z^5+5 a^3 z^3+a^2 z^6-5 a^2 z^4+6 a^2 z^2-2 a^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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                     1 1 -1                       0 -3               1 2 1   0 -5             1 1 1     1 -7           2 4 1       1 -9         1 1 2         2 -11       1 3 1           1 -13     1 1 1             1 -15     1 1               0 -17 1 1                   0 -19 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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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