L11a260

L11a259

L11a261

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Link Presentations

[edit Notes on L11a260's Link Presentations]

Planar diagram presentation	X10,1,11,2 X14,5,15,6 X12,3,13,4 X18,8,19,7 X22,15,9,16 X20,17,21,18 X16,21,17,22 X4,13,5,14 X6,20,7,19 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 3, -8, 2, -9, 4, -11}, {10, -1, 11, -3, 8, -2, 5, -7, 6, -4, 9, -6, 7, -5}

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