L11n222

L11n221

L11n223

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Link Presentations

[edit Notes on L11n222's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X13,20,14,21 X5,14,6,15 X4,21,5,22 X16,9,17,10 X22,15,9,16 X17,6,18,7 X7,18,8,19 X19,8,20,1
Gauss code	{1, -2, 3, -6, -5, 9, -10, 11}, {7, -1, 2, -3, -4, 5, 8, -7, -9, 10, -11, 4, 6, -8}

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{(u v+1) \left(u^2 v^4-u v^3+2 u v^2-u v+1\right)}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}-\frac{1}{q^{27/2}}+\frac{2}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} \left(-z^3\right)-2 a^{13} z-a^{13} z^{-1} +a^{11} z^7+8 a^{11} z^5+19 a^{11} z^3+15 a^{11} z+3 a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-36 a^9 z^3-17 a^9 z-2 a^9 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^{17}+z a^{17}-2 z^4 a^{16}+3 z^2 a^{16}-z^5 a^{15}+z a^{15}-2 z^4 a^{14}+2 z^2 a^{14}-a^{14}-z^3 a^{13}-z a^{13}+a^{13} z^{-1} -z^8 a^{12}+8 z^6 a^{12}-19 z^4 a^{12}+14 z^2 a^{12}-3 a^{12}-z^9 a^{11}+9 z^7 a^{11}-27 z^5 a^{11}+34 z^3 a^{11}-18 z a^{11}+3 a^{11} z^{-1} -z^8 a^{10}+8 z^6 a^{10}-19 z^4 a^{10}+15 z^2 a^{10}-3 a^{10}-z^9 a^9+9 z^7 a^9-28 z^5 a^9+36 z^3 a^9-17 z a^9+2 a^9 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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                     1 1 -10                     1 1 -12                 1     1 -14             1         1 -16           1 1 1       -1 -18         2 1           1 -20         1 1           0 -22     2 2               0 -24   1 1                 0 -26   1                   -1 -28 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-10 }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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