L11a256

L11a255

L11a257

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Link Presentations

[edit Notes on L11a256's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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^3 v^4-2 u^3 v^3+2 u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+7 u^2 v^3-8 u^2 v^2+4 u^2 v-u^2-u v^5+4 u v^4-8 u v^3+7 u v^2-4 u v+u-v^4+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{12}{q^{9/2}}-q^{7/2}+\frac{16}{q^{7/2}}+4 q^{5/2}-\frac{20}{q^{5/2}}-9 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+14 \sqrt{q}-\frac{18}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+a z^7-a^5 z^5+4 a^3 z^5+3 a z^5-z^5 a^{-1} -3 a^5 z^3+6 a^3 z^3+2 a z^3-2 z^3 a^{-1} -2 a^5 z+2 a^3 z-a z-z a^{-1} +a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^4 z^{10}-2 a^2 z^{10}-4 a^5 z^9-10 a^3 z^9-6 a z^9-4 a^6 z^8-3 a^4 z^8-8 a^2 z^8-9 z^8-3 a^7 z^7+6 a^5 z^7+21 a^3 z^7+4 a z^7-8 z^7 a^{-1} -a^8 z^6+8 a^6 z^6+9 a^4 z^6+18 a^2 z^6-4 z^6 a^{-2} +14 z^6+9 a^7 z^5-a^5 z^5-22 a^3 z^5+2 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-2 a^6 z^4-4 a^4 z^4-11 a^2 z^4+5 z^4 a^{-2} -7 z^4-8 a^7 z^3-3 a^5 z^3+13 a^3 z^3+a z^3-6 z^3 a^{-1} +z^3 a^{-3} -2 a^8 z^2-2 a^6 z^2-a^4 z^2+2 a^2 z^2-z^2 a^{-2} +2 z^2+2 a^7 z+4 a^5 z-a^3 z-2 a z+z a^{-1} +a^4-a^5 z^{-1} -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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 3     -5 0               10 6       4 -2             11 9         -2 -4           9 9           0 -6         7 11             4 -8       5 9               -4 -10     2 8                 6 -12   1 4                   -3 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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