L11n175

L11n174

L11n176

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Link Presentations

[edit Notes on L11n175's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X14,5,15,6 X18,12,19,11 X3,10,4,11 X12,7,13,8 X16,13,17,14 X22,17,7,18 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {6, -1, -2, 5, 4, -6, 7, -3, 9, -7, 8, -4, -11, 2, 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{-3 u^2 v^2+3 u^2 v-u^2+2 u v^4-5 u v^3+5 u v^2-5 u v+2 u-v^4+3 v^3-3 v^2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{4}{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^{-1} +a^7 z^5+a^7 z^3-a^7 z-2 a^7 z^{-1} +a^5 z^5+a^5 z^3+a^5 z+2 a^5 z^{-1} -2 a^3 z^3-3 a^3 z-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-2 a^{12} z^4+4 a^{11} z^7-12 a^{11} z^5+7 a^{11} z^3+a^{11} z+5 a^{10} z^8-15 a^{10} z^6+11 a^{10} z^4-3 a^{10} z^2+2 a^9 z^9+2 a^9 z^7-19 a^9 z^5+17 a^9 z^3-5 a^9 z+a^9 z^{-1} +9 a^8 z^8-27 a^8 z^6+27 a^8 z^4-11 a^8 z^2+2 a^7 z^9+a^7 z^7-12 a^7 z^5+17 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-10 a^6 z^6+16 a^6 z^4-9 a^6 z^2+a^6+3 a^5 z^7-5 a^5 z^5+10 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^6+2 a^4 z^4-a^4 z^2+3 a^3 z^3-4 a^3 z+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 1 -2 -6               5 1   4 -8             5 3     -2 -10           6 5       1 -12         5 5         0 -14       4 6           -2 -16     3 6             3 -18   1 3               -2 -20   3                 3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\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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