L11n84

L11n83

L11n85

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Link Presentations

[edit Notes on L11n84's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X9,18,10,19 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,8,18,9 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 8, -4, -9, 11, -2, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}

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(2)^3-t(2)^3-3 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-3 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+4 q^{5/2}-5 q^{3/2}+6 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-1} -2 z a^3-a^3 z^{-1} +2 z^3 a+2 z a-z^5 a^{-1} -3 z^3 a^{-1} -3 z a^{-1} +z^3 a^{-3} +z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^3-2 a^5 z+a^5 z^{-1} +z^6 a^{-4} +a^4 z^4-3 z^4 a^{-4} +z^2 a^{-4} -a^4+3 z^7 a^{-3} +2 a^3 z^5-11 z^5 a^{-3} -2 a^3 z^3+8 z^3 a^{-3} -z a^{-3} +a^3 z^{-1} +a^2 z^8+3 z^8 a^{-2} -4 a^2 z^6-11 z^6 a^{-2} +9 a^2 z^4+9 z^4 a^{-2} -5 a^2 z^2-2 z^2 a^{-2} +a z^9+z^9 a^{-1} -3 a z^7+3 a z^5-10 z^5 a^{-1} +11 z^3 a^{-1} +a z-2 z a^{-1} +4 z^8-16 z^6+20 z^4-8 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               2 1   -1 4             3 2     1 2           3 2       -1 0         3 3         0 -2       3 4           1 -4     1 2             -1 -6     3               3 -8 1 1                 0 -10 1                   1

Integral Khovanov Homology (db, data source)

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