L11n123

L11n122

L11n124

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Link Presentations

[edit Notes on L11n123's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X9,16,10,17 X20,11,21,12 X8,21,9,22 X18,7,19,8 X12,19,13,20 X15,10,16,11 X22,17,5,18 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -5, -3, 8, 4, -7, 11, -2, -8, 3, 9, -6, 7, -4, 5, -9}

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)^5+3 t(1) t(2)^4-3 t(2)^4-4 t(1) t(2)^3+4 t(2)^3+4 t(1) t(2)^2-4 t(2)^2-3 t(1) t(2)+3 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{11}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{9}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{11}-2 a^{11} z^{-1} +4 z^3 a^9+10 z a^9+5 a^9 z^{-1} -3 z^5 a^7-11 z^3 a^7-11 z a^7-3 a^7 z^{-1} -z^5 a^5-2 z^3 a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{14}+2 z^2 a^{14}-a^{14}-3 z^5 a^{13}+3 z^3 a^{13}-z^8 a^{12}+z^6 a^{12}-6 z^4 a^{12}+6 z^2 a^{12}-z^9 a^{11}-z^5 a^{11}-2 z^3 a^{11}+4 z a^{11}-2 a^{11} z^{-1} -5 z^8 a^{10}+10 z^6 a^{10}-8 z^4 a^{10}-4 z^2 a^{10}+5 a^{10}-z^9 a^9-6 z^7 a^9+20 z^5 a^9-25 z^3 a^9+15 z a^9-5 a^9 z^{-1} -4 z^8 a^8+6 z^6 a^8+2 z^4 a^8-8 z^2 a^8+5 a^8-6 z^7 a^7+17 z^5 a^7-18 z^3 a^7+11 z a^7-3 a^7 z^{-1} -3 z^6 a^6+5 z^4 a^6-z^5 a^5+2 z^3 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 3 1 -2 -8               4     4 -10             4 3     -1 -12           7 4       3 -14         4 5         1 -16       5 6           -1 -18     2 4             2 -20   1 5               -4 -22   2                 2 -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{ 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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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} }[/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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