L11n209

L11n208

L11n210

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Link Presentations

[edit Notes on L11n209's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X19,5,20,4 X6,22,7,21 X16,7,17,8 X8,9,1,10 X13,18,14,19 X20,15,21,16
Gauss code	{1, -2, 3, 6, -4, -7, 8, -9}, {9, -1, 2, -3, -10, 4, 11, -8, 5, 10, -6, -11, 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)+1) (t(2) t(1)-t(1)+1) (t(1) t(2)-t(2)+1)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^5+4 a^7 z^3+4 a^7 z-a^5 z^7-6 a^5 z^5-12 a^5 z^3-7 a^5 z+a^5 z^{-1} +a^3 z^5+3 a^3 z^3+a^3 z-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-3 a^{11} z^3+a^{11} z+2 a^{10} z^6-6 a^{10} z^4+3 a^{10} z^2+2 a^9 z^7-5 a^9 z^5+2 a^9 z^3+2 a^8 z^8-7 a^8 z^6+10 a^8 z^4-5 a^8 z^2+a^7 z^9-3 a^7 z^7+5 a^7 z^5-3 a^7 z^3+3 a^7 z+3 a^6 z^8-13 a^6 z^6+24 a^6 z^4-12 a^6 z^2+a^5 z^9-5 a^5 z^7+14 a^5 z^5-15 a^5 z^3+6 a^5 z+a^5 z^{-1} +a^4 z^8-4 a^4 z^6+9 a^4 z^4-6 a^4 z^2-a^4+3 a^3 z^5-7 a^3 z^3+2 a^3 z+a^3 z^{-1} +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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 0                   1 -1 -2                 2   2 -4               2 2   0 -6             4 1     3 -8           2 2       0 -10         4 4         0 -12       2 3           1 -14     1 3             -2 -16   1 2               1 -18   1                 -1 -20 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/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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