L11n18

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L11n19

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Link Presentations

[edit Notes on L11n18's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X9,16,10,17 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 10, -7, 4, 11, -2, 3, 6, -8, 9, -10, 7, -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{(u-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)-a^9 z-a^9 z^{-1} +a^7 z^5+2 a^7 z^3+2 a^7 z+a^7 z^{-1} +a^5 z^5+2 a^5 z^3+3 a^5 z+2 a^5 z^{-1} -2 a^3 z^3-4 a^3 z-2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+3 z^4 a^{12}-3 z^2 a^{12}+a^{12}-3 z^7 a^{11}+9 z^5 a^{11}-6 z^3 a^{11}+z a^{11}-3 z^8 a^{10}+6 z^6 a^{10}+3 z^4 a^{10}-5 z^2 a^{10}-z^9 a^9-4 z^7 a^9+17 z^5 a^9-12 z^3 a^9+2 z a^9+a^9 z^{-1} -5 z^8 a^8+10 z^6 a^8-3 z^4 a^8+2 z^2 a^8-3 a^8-z^9 a^7-3 z^7 a^7+11 z^5 a^7-11 z^3 a^7+2 z a^7+a^7 z^{-1} -2 z^8 a^6+2 z^6 a^6-3 z^4 a^6+z^2 a^6-2 z^7 a^5+3 z^5 a^5-8 z^3 a^5+6 z a^5-2 a^5 z^{-1} -z^6 a^4-3 z^2 a^4+3 a^4-3 z^3 a^3+5 z a^3-2 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               4     4 -8             4 3     -1 -10           6 4       2 -12         3 4         1 -14       4 6           -2 -16     2 3             1 -18   1 4               -3 -20   2                 2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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