L11a230

L11a229

L11a231

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Link Presentations

[edit Notes on L11a230's Link Presentations]

Planar diagram presentation	X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X6718 X4,15,5,16 X22,20,7,19 X20,10,21,9 X10,22,11,21
Gauss code	{1, -2, 3, -8, 4, -7}, {7, -1, 10, -11, 2, -3, 5, -6, 8, -4, 6, -5, 9, -10, 11, -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{2 u^2 v^4-5 u^2 v^3+6 u^2 v^2-2 u^2 v-u v^4+7 u v^3-11 u v^2+7 u v-u-2 v^3+6 v^2-5 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-3 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+3 z^3 a^7+3 z a^7-z^7 a^5-4 z^5 a^5-6 z^3 a^5-2 z a^5+2 a^5 z^{-1} -z^7 a^3-4 z^5 a^3-7 z^3 a^3-7 z a^3-3 a^3 z^{-1} +z^5 a+3 z^3 a+3 z a+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+7 z^5 a^9-3 z^3 a^9+z a^9-6 z^8 a^8+9 z^6 a^8-8 z^4 a^8+4 z^2 a^8-4 z^9 a^7+6 z^5 a^7-4 z^3 a^7-z a^7-z^{10} a^6-11 z^8 a^6+26 z^6 a^6-24 z^4 a^6+9 z^2 a^6-7 z^9 a^5+6 z^7 a^5+8 z^5 a^5-10 z^3 a^5+4 z a^5-2 a^5 z^{-1} -z^{10} a^4-9 z^8 a^4+21 z^6 a^4-10 z^4 a^4-2 z^2 a^4+3 a^4-3 z^9 a^3-2 z^7 a^3+18 z^5 a^3-18 z^3 a^3+10 z a^3-3 a^3 z^{-1} -4 z^8 a^2+6 z^6 a^2+4 z^4 a^2-8 z^2 a^2+3 a^2-3 z^7 a+8 z^5 a-7 z^3 a+3 z a-a z^{-1} -z^6+3 z^4-3 z^2+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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   5 1   -4 -2                 7 2     5 -4               9 6       -3 -6             10 6         4 -8           8 9           1 -10         8 10             -2 -12       4 8               4 -14     3 8                 -5 -16   1 5                   4 -18   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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