L11n150

L11n149

L11n151

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Link Presentations

[edit Notes on L11n150's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X15,5,16,4 X11,16,12,17 X20,13,21,14 X14,21,15,22 X2,18,3,17
Gauss code	{1, -11, -6, 7, 5, -3}, {3, -1, 2, 6, -8, -5, 9, -10, -7, 8, 11, -2, 4, -9, 10, -4}

A Braid Representative	{{{braid_table}}}
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^2 v^2-u^2 v-3 u v^2+5 u v-3 u-v+1}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{3}{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-a^9 z^{-1} +z^5 a^7+4 z^3 a^7+6 z a^7+2 a^7 z^{-1} -2 z^3 a^5-3 z a^5-z^3 a^3-2 z a^3-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+4 z^4 a^{12}-3 z^2 a^{12}-2 z^7 a^{11}+8 z^5 a^{11}-7 z^3 a^{11}+z a^{11}-2 z^8 a^{10}+8 z^6 a^{10}-9 z^4 a^{10}+6 z^2 a^{10}-2 a^{10}-z^9 a^9+3 z^7 a^9-2 z^5 a^9+2 z^3 a^9-2 z a^9+a^9 z^{-1} -3 z^8 a^8+14 z^6 a^8-25 z^4 a^8+20 z^2 a^8-5 a^8-z^9 a^7+5 z^7 a^7-12 z^5 a^7+13 z^3 a^7-8 z a^7+2 a^7 z^{-1} -z^8 a^6+5 z^6 a^6-13 z^4 a^6+11 z^2 a^6-3 a^6-2 z^5 a^5+3 z^3 a^5-3 z a^5-z^4 a^4+a^4-z^3 a^3+2 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                   1 1 -4                 1 1 0 -6               3     3 -8             2 2     0 -10           3 2       1 -12         2 2         0 -14       2 3           -1 -16     1 2             1 -18   1 2               -1 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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