L11a368

L11a367

L11a369

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Link Presentations

[edit Notes on L11a368's Link Presentations]

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

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^4 v^3-u^4 v^2+u^3 v^4-3 u^3 v^3+4 u^3 v^2-u^3 v-u^2 v^4+4 u^2 v^3-5 u^2 v^2+4 u^2 v-u^2-u v^3+4 u v^2-3 u v+u-v^2+v}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-3 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+4 z^3 a^7+4 z a^7+a^7 z^{-1} -z^7 a^5-5 z^5 a^5-9 z^3 a^5-7 z a^5-a^5 z^{-1} -z^7 a^3-4 z^5 a^3-4 z^3 a^3-z a^3+z^5 a+3 z^3 a+z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-3 a^{11} z^3+2 a^{11} z+2 a^{10} z^6-5 a^{10} z^4+3 a^{10} z^2+2 a^9 z^7-2 a^9 z^5-2 a^9 z^3+a^9 z+2 a^8 z^8-2 a^8 z^6+a^8 z^4-a^8 z^2+2 a^7 z^9-5 a^7 z^7+13 a^7 z^5-15 a^7 z^3+7 a^7 z-a^7 z^{-1} +a^6 z^{10}-3 a^6 z^6+10 a^6 z^4-7 a^6 z^2+a^6+5 a^5 z^9-18 a^5 z^7+31 a^5 z^5-24 a^5 z^3+9 a^5 z-a^5 z^{-1} +a^4 z^{10}+2 a^4 z^8-13 a^4 z^6+17 a^4 z^4-6 a^4 z^2+3 a^3 z^9-8 a^3 z^7+5 a^3 z^5-a^3 z^3+4 a^2 z^8-13 a^2 z^6+10 a^2 z^4-2 a^2 z^2+3 a z^7-10 a z^5+7 a z^3-a z+z^6-3 z^4+z^2 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   3 1   -2 -2                 5 2     3 -4               6 4       -2 -6             6 4         2 -8           5 6           1 -10         5 6             -1 -12       2 5               3 -14     2 5                 -3 -16   1 3                   2 -18   1                     -1 -20 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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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