L11n252

L11n251

L11n253

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Link Presentations

[edit Notes on L11n252's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,3,15,4 X9,18,10,19 X5,16,6,17 X22,7,11,8 X6,21,7,22 X15,21,16,20 X17,8,18,9 X19,4,20,5 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 2, 9, -4, -6, 5, 8, -3, -11}, {10, -1, 11, -2, -7, 4, -8, 3, -9, 7, 6, -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(2)^2 t(1)^4-t(2)^3 t(1)^3+t(2)^2 t(1)^3-t(1)^3-t(2)^2 t(1)^2-t(2)^4 t(1)+t(2)^2 t(1)-t(2) t(1)-t(2)^2}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}-\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z+a^{11} z^{-1} -a^9 z^{-1} -a^7 z^5-4 a^7 z^3-3 a^7 z-a^5 z^5-4 a^5 z^3-3 a^5 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^{13}+7 z^5 a^{13}-14 z^3 a^{13}+8 z a^{13}+z^4 a^{12}-2 z^2 a^{12}+z^5 a^{11}-3 z^3 a^{11}+a^{11} z^{-1} -z^2 a^{10}-a^{10}-z^5 a^9+2 z^3 a^9-2 z a^9+a^9 z^{-1} -z^6 a^8+2 z^4 a^8-z^7 a^7+4 z^5 a^7-5 z^3 a^7+3 z a^7-z^6 a^6+3 z^4 a^6-z^2 a^6-z^5 a^5+4 z^3 a^5-3 z a^5 }[/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 -4                     1 1 -6                   1 1 0 -8                 1     1 -10               1 1     0 -12           1 2 1       0 -14           1 1         0 -16       1 2 2           -1 -18         1             1 -20     1 1               0 -22 1                     1 -24 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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