L11n299

L11n298

L11n300

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Link Presentations

[edit Notes on L11n299's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,14,10,15 X15,20,16,21 X19,5,20,10 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -5, 6, -7, 9}, {11, -2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}

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(2)^2 t(3)^4+t(2) t(3)^4+2 t(2)^2 t(3)^3-t(2) t(3)^3-t(3)^3-t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2+t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2+t(3)^2+t(1) t(2)^2 t(3)-2 t(1) t(3)+t(1) t(2) t(3)+t(1)-t(1) t(2)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-7} +3 q^{-6} -3 q^{-5} +4 q^{-4} +q^3-2 q^{-3} -2 q^2+3 q^{-2} +q }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6+a^6 z^{-2} -a^6+2 z^4 a^4+6 z^2 a^4-2 a^4 z^{-2} +a^4-z^6 a^2-5 z^4 a^2-6 z^2 a^2+a^2 z^{-2} -a^2+z^2+1+z^2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^5 z^9+2 a^3 z^9+3 a^6 z^8+6 a^4 z^8+4 a^2 z^8+z^8+a^7 z^7-8 a^5 z^7-9 a^3 z^7+2 a z^7+2 z^7 a^{-1} -15 a^6 z^6-35 a^4 z^6-27 a^2 z^6+z^6 a^{-2} -6 z^6-4 a^7 z^5+a^5 z^5-15 a z^5-10 z^5 a^{-1} +19 a^6 z^4+55 a^4 z^4+48 a^2 z^4-4 z^4 a^{-2} +8 z^4+3 a^7 z^3+10 a^5 z^3+18 a^3 z^3+21 a z^3+10 z^3 a^{-1} -8 a^6 z^2-29 a^4 z^2-30 a^2 z^2+2 z^2 a^{-2} -7 z^2-a^7 z-3 a^5 z-7 a^3 z-7 a z-2 z a^{-1} +3 a^4+4 a^2+2-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     1   -1 3                 1 1 1   -1 1               2 2 1     1 -1             2 3 1       0 -3           3 3 3         3 -5         3 5 1           1 -7       2 2 2             2 -9     2 4 1               1 -11   1 1                   0 -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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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}\oplus{\mathbb Z}_2 }[/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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