L11n44

L11n43

L11n45

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Link Presentations

[edit Notes on L11n44's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,14,6,15 X3849 X9,18,10,19 X11,20,12,21 X13,22,14,5 X19,10,20,11 X21,12,22,13 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, -11, -2, 3, 6, -9, 7, -10, 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^7-1}{\sqrt{u} v^{7/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{27/2}} }[/math] (db)
Signature	-9 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{13}-4 z a^{13}-3 a^{13} z^{-1} +z^7 a^{11}+8 z^5 a^{11}+21 z^3 a^{11}+21 z a^{11}+7 a^{11} z^{-1} -z^9 a^9-9 z^7 a^9-28 z^5 a^9-36 z^3 a^9-19 z a^9-4 a^9 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{18}+a^{13} z^5-6 a^{13} z^3+9 a^{13} z-3 a^{13} z^{-1} +a^{12} z^8-8 a^{12} z^6+21 a^{12} z^4-21 a^{12} z^2+7 a^{12}+a^{11} z^9-9 a^{11} z^7+29 a^{11} z^5-42 a^{11} z^3+28 a^{11} z-7 a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+21 a^{10} z^4-21 a^{10} z^2+7 a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-36 a^9 z^3+19 a^9 z-4 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                   1 1 -10                   1 1 -12               1     1 -14           1         1 -16           2 1       1 -18       1             1 -20       1 1           0 -22   1 1               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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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