L11n264

L11n263

L11n265

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Link Presentations

[edit Notes on L11n264's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,12,19,11 X19,22,20,9 X15,20,16,21 X21,16,22,17 X12,18,13,17 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+t(1) t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2-t(3)^2-t(1) t(3)+t(1) t(2) t(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{ 1-3 q^{-1} +5 q^{-2} -5 q^{-3} +7 q^{-4} -5 q^{-5} +6 q^{-6} -3 q^{-7} + q^{-8} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^{-2} +z^4 a^6+z^2 a^6-2 a^6 z^{-2} -3 a^6-z^6 a^4-3 z^4 a^4+a^4 z^{-2} +3 a^4+z^4 a^2+2 z^2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^9 z^3+a^8 z^8-5 a^8 z^6+11 a^8 z^4-6 a^8 z^2-a^8 z^{-2} +3 a^8+a^7 z^9-3 a^7 z^7+2 a^7 z^5+3 a^7 z^3-3 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-16 a^6 z^6+22 a^6 z^4-14 a^6 z^2-2 a^6 z^{-2} +5 a^6+a^5 z^9-8 a^5 z^5+7 a^5 z^3-3 a^5 z+2 a^5 z^{-1} +3 a^4 z^8-10 a^4 z^6+8 a^4 z^4-6 a^4 z^2-a^4 z^{-2} +3 a^4+3 a^3 z^7-10 a^3 z^5+6 a^3 z^3+a^2 z^6-3 a^2 z^4+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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 1                 1 1 -1               2   -2 -3             3 1   2 -5           3 3     0 -7         4 2       2 -9     1 2 3         2 -11     5 4           1 -13   1 4             3 -15   2               -2 -17 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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