L11n78

L11n77

L11n79

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Link Presentations

[edit Notes on L11n78's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X19,1,20,4 X5,12,6,13 X3,10,4,11 X13,22,14,5 X21,14,22,15 X11,20,12,21 X2,16,3,15
Gauss code	{1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -10, 6, -8, 9, 11, -2, 4, -3, -5, 10, -9, 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{2 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9-2 z a^9-2 a^9 z^{-1} +z^5 a^7+3 z^3 a^7+5 z a^7+4 a^7 z^{-1} +z^5 a^5+2 z^3 a^5+z a^5-a^5 z^{-1} -2 z^3 a^3-4 z a^3-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+4 z^4 a^{12}-5 z^2 a^{12}+2 a^{12}-2 z^7 a^{11}+6 z^5 a^{11}-3 z^3 a^{11}-z a^{11}-2 z^8 a^{10}+4 z^6 a^{10}+2 z^4 a^{10}-3 z^2 a^{10}+a^{10}-z^9 a^9+3 z^5 a^9+4 z^3 a^9-6 z a^9+2 a^9 z^{-1} -4 z^8 a^8+11 z^6 a^8-14 z^4 a^8+15 z^2 a^8-6 a^8-z^9 a^7+z^5 a^7+3 z^3 a^7-9 z a^7+4 a^7 z^{-1} -2 z^8 a^6+5 z^6 a^6-12 z^4 a^6+12 z^2 a^6-5 a^6-2 z^7 a^5+4 z^5 a^5-7 z^3 a^5+z a^5+a^5 z^{-1} -z^6 a^4-z^2 a^4+a^4-3 z^3 a^3+5 z a^3-a^3 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 -2                   2 2 -4                 3 2 -1 -6               3     3 -8             4 3     -1 -10           5 3       2 -12         2 4         2 -14       4 5           -1 -16     1 2             1 -18   1 4               -3 -20   1                 1 -22 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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