L11n397

L11n396

L11n398

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Link Presentations

[edit Notes on L11n397's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X17,22,18,19 X13,20,14,21 X19,14,20,15 X21,18,22,5 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {-7, 6, -8, 5}, {10, -1, 3, -9, 4, -2, 11, -3, -6, 7, 9, -4, -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-1) (w-1)^2 (w+1) \left(v+w^2\right)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+2 q^4-2 q^3+q^2-q+1+2 q^{-1} - q^{-2} +3 q^{-3} - q^{-4} + q^{-5} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6-2 a^2 z^4+6 z^4+a^4 z^2-9 a^2 z^2-2 z^2 a^{-2} -z^2 a^{-4} +11 z^2+3 a^4-11 a^2-3 a^{-2} +11+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-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^8-5 a^3 z^7-4 a z^7+2 z^7 a^{-1} +z^7 a^{-3} -7 a^4 z^6-27 a^2 z^6+2 z^6 a^{-4} -22 z^6+3 a^3 z^5-7 a z^5-15 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} +17 a^4 z^4+56 a^2 z^4-7 z^4 a^{-4} +46 z^4+11 a^3 z^3+32 a z^3+27 z^3 a^{-1} +3 z^3 a^{-3} -3 z^3 a^{-5} -19 a^4 z^2-49 a^2 z^2-5 z^2 a^{-2} +3 z^2 a^{-4} -38 z^2-15 a^3 z-29 a z-17 z a^{-1} -2 z a^{-3} +z a^{-5} +10 a^4+22 a^2+4 a^{-2} +17+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 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 4 5 χ 11                       1 -1 9                     1   1 7                   1 1   0 5               2 2 1     -1 3             1 2 1       0 1           2 4 2         0 -1         1 2 4           3 -3       1 2               1 -5     2 1 1               2 -7   1 3                   2 -9                         0 -11 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{ i=3 }[/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}^{3} }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{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=4 }[/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 }[/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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