L11n393

L11n392

L11n394

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Link Presentations

[edit Notes on L11n393's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X18,12,5,11 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {4, -1, 2, -5, -7, 8, 10, -4, -6, 9, 11, -2, 3, -10}

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{(t(1)-1) (t(3)-1)^2 \left(t(3)^3+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-7} +3 q^{-6} -4 q^{-5} +7 q^{-4} +q^3-5 q^{-3} -q^2+6 q^{-2} -q-4 q^{-1} +3 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^2\right)+a^6 z^{-2} -a^6+2 a^4 z^4+5 a^4 z^2-2 a^4 z^{-2} +a^4-a^2 z^6-4 a^2 z^4-4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} -z^2-1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-4 a^7 z^5+4 a^7 z^3-a^7 z+3 a^6 z^8-14 a^6 z^6+19 a^6 z^4-10 a^6 z^2+a^6 z^{-2} +2 a^5 z^9-5 a^5 z^7-7 a^5 z^5+14 a^5 z^3-3 a^5 z-2 a^5 z^{-1} +8 a^4 z^8-39 a^4 z^6+55 a^4 z^4-31 a^4 z^2+2 a^4 z^{-2} +4 a^4+2 a^3 z^9-4 a^3 z^7-15 a^3 z^5+30 a^3 z^3-11 a^3 z-2 a^3 z^{-1} +5 a^2 z^8-27 a^2 z^6+z^6 a^{-2} +41 a^2 z^4-5 z^4 a^{-2} -23 a^2 z^2+4 z^2 a^{-2} +a^2 z^{-2} +5 a^2- a^{-2} +3 a z^7+z^7 a^{-1} -19 a z^5-7 z^5 a^{-1} +30 a z^3+10 z^3 a^{-1} -13 a z-4 z a^{-1} -z^6+2 z^2+1 }[/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                 2 1 1   -2 1               3 1       2 -1             4 4 1       -1 -3           3 3 2         2 -5         3 4             1 -7       4 3 1             2 -9     2 5                 3 -11   1 2                   -1 -13   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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 }[/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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