L11n263

L11n262

L11n264

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Link Presentations

[edit Notes on L11n263's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X22,20,9,19 X20,16,21,15 X16,22,17,21 X17,12,18,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, 7, -8, -9, 5, 6, -7, 8, -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{-t(1) t(3)^3+t(1) t(2) t(3)^3-t(2) t(3)^3+t(3)^3+3 t(1) t(3)^2-2 t(1) t(2) t(3)^2+3 t(2) t(3)^2-3 t(3)^2-3 t(1) t(3)+3 t(1) t(2) t(3)-3 t(2) t(3)+2 t(3)+t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-7} -2 q^{-6} +7 q^{-5} -8 q^{-4} +11 q^{-3} -q^2-10 q^{-2} +4 q+9 q^{-1} -7 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} -2 a^6 z^{-2} -3 a^6-a^4 z^4+a^4 z^2+a^4 z^{-2} +3 a^4+a^2 z^6+3 a^2 z^4+3 a^2 z^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^9+a^3 z^9+a^6 z^8+5 a^4 z^8+4 a^2 z^8-a^5 z^7+5 a^3 z^7+6 a z^7-3 a^6 z^6-12 a^4 z^6-5 a^2 z^6+4 z^6+2 a^7 z^5+3 a^5 z^5-12 a^3 z^5-12 a z^5+z^5 a^{-1} +a^8 z^4+12 a^6 z^4+17 a^4 z^4-a^2 z^4-7 z^4-a^7 z^3+2 a^5 z^3+7 a^3 z^3+3 a z^3-z^3 a^{-1} -3 a^8 z^2-12 a^6 z^2-10 a^4 z^2+z^2-3 a^7 z-3 a^5 z+3 a^8+5 a^6+3 a^4+2 a^7 z^{-1} +2 a^5 z^{-1} -a^8 z^{-2} -2 a^6 z^{-2} -a^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 χ 5                   1 -1 3                 3   3 1               4 1   -3 -1             5 3     2 -3           6 5       -1 -5         5 4         1 -7       3 6           3 -9     4 5             -1 -11   1 6               5 -13   1                 -1 -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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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