L11n234

L11n233

L11n235

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Link Presentations

[edit Notes on L11n234's Link Presentations]

Planar diagram presentation	X10,1,11,2 X5,20,6,21 X3,15,4,14 X15,5,16,4 X19,22,20,9 X7,17,8,16 X11,18,12,19 X17,12,18,13 X2,9,3,10 X13,1,14,8 X21,6,22,7
Gauss code	{1, -9, -3, 4, -2, 11, -6, 10}, {9, -1, -7, 8, -10, 3, -4, 6, -8, 7, -5, 2, -11, 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)-1) (t(2)-1) \left(t(1)^2 t(2)^2-t(1) t(2)^2+5 t(1) t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}-q^{7/2}+\frac{8}{q^{7/2}}+3 q^{5/2}-\frac{12}{q^{5/2}}-7 q^{3/2}+\frac{12}{q^{3/2}}+\frac{2}{q^{11/2}}+10 \sqrt{q}-\frac{12}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z-a^3 z^5-2 a^3 z^3-2 a^3 z+a z^7+5 a z^5-z^5 a^{-1} +10 a z^3-3 z^3 a^{-1} +7 a z-4 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-2 a^4 z^8-6 a^2 z^8-4 z^8-a^5 z^7-5 a^3 z^7-9 a z^7-5 z^7 a^{-1} +6 a^2 z^6-3 z^6 a^{-2} +3 z^6-3 a^5 z^5+9 a^3 z^5+22 a z^5+9 z^5 a^{-1} -z^5 a^{-3} -3 a^6 z^4+a^2 z^4+5 z^4 a^{-2} +3 z^4+6 a^5 z^3-5 a^3 z^3-19 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} +4 a^6 z^2+a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -z^2-3 a^5 z+8 a z+4 z a^{-1} -z a^{-3} +1-a z^{-1} - a^{-1} z^{-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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   1 1 6                 2   -2 4               5 1   4 2             5 2     -3 0           7 5       2 -2         7 7         0 -4       5 5           0 -6     3 7             4 -8   2 5               -3 -10   3                 3 -12 2                   -2

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=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/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}_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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