L11n53

L11n52

L11n54

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Link Presentations

[edit Notes on L11n53's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X13,20,14,21 X17,5,18,22 X21,19,22,18 X19,14,20,15 X15,8,16,9 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 9, -4, -2, 11, -3, -5, 8, -9, 4, -6, 7, -8, 5, -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{(t(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+\frac{1}{q^{7/2}}+4 q^{5/2}-\frac{3}{q^{5/2}}-5 q^{3/2}+\frac{4}{q^{3/2}}+5 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} +2 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -a^3 z+3 a z-3 z a^{-1} +z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-4} -3 z^4 a^{-4} +z^2 a^{-4} +3 z^7 a^{-3} -11 z^5 a^{-3} +2 a^3 z^3+8 z^3 a^{-3} -2 a^3 z-2 z a^{-3} +a^2 z^8+3 z^8 a^{-2} -5 a^2 z^6-11 z^6 a^{-2} +9 a^2 z^4+8 z^4 a^{-2} -3 a^2 z^2-z^2 a^{-2} +a z^9+z^9 a^{-1} -3 a z^7-a z^5-12 z^5 a^{-1} +10 a z^3+16 z^3 a^{-1} -6 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +4 z^8-17 z^6+20 z^4-5 z^2-1 }[/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   \ -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       1 3 2       0 0       4 3         1 -2     2 4           2 -4   1 2             -1 -6   2               2 -8 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{ i=2 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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