L11n272

L11n271

L11n273

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Link Presentations

[edit Notes on L11n272's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X18,11,19,12 X22,17,9,18 X12,21,13,22 X20,13,21,14 X14,19,15,20 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -7, 8, -9, -4, 3, 6, -5, 9, -8, 7, -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{2 t(3)^3-2 t(1) t(3)^2-2 t(2) t(3)^2-t(3)^2+2 t(1) t(3)+t(1) t(2) t(3)+2 t(2) t(3)-2 t(1) t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-q^2+2 q+1+ q^{-1} + q^{-2} - q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} - q^{-7} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^2\right)-a^6 z^{-2} -2 a^6+a^4 z^4+4 a^4 z^2+3 a^4 z^{-2} +6 a^4-a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} -3 a^2+2 a^{-2} -z^4-4 z^2- z^{-2} -3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-6 a^7 z^5+11 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +a^6 z^8-5 a^6 z^6+6 a^6 z^4-2 a^6 z^2-a^6 z^{-2} +a^6+3 a^5 z^7-18 a^5 z^5+33 a^5 z^3-27 a^5 z+8 a^5 z^{-1} +a^4 z^8-5 a^4 z^6+5 a^4 z^4-4 a^4 z^2-3 a^4 z^{-2} +5 a^4+3 a^3 z^7-21 a^3 z^5+42 a^3 z^3-34 a^3 z+10 a^3 z^{-1} +a^2 z^8-6 a^2 z^6+z^6 a^{-2} +7 a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2+6 z^2 a^{-2} -2 a^2 z^{-2} + a^{-2} z^{-2} +4 a^2-4 a^{-2} +2 a z^7+z^7 a^{-1} -13 a z^5-4 z^5 a^{-1} +20 a z^3-10 a z+4 z a^{-1} +2 a z^{-1} -2 a^{-1} z^{-1} +z^8-5 z^6+3 z^4+5 z^2+ z^{-2} -3 }[/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                     1 1 0 3                   1     1 1               3 1 1     3 -1             2 5 1       2 -3           1   1         2 -5         1 3 2           0 -7       2 1               1 -9       1 1               0 -11   1 2                   -1 -13                         0 -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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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