L10a11

L10a10

L10a12

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Link Presentations

[edit Notes on L10a11's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X14,7,15,8 X8,15,9,16 X20,17,5,18 X18,14,19,13 X12,20,13,19 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, 3, -8, 7, -4, 5, -3, 6, -7, 8, -6}

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{2 u v^4-7 u v^3+7 u v^2-4 u v+u+v^5-4 v^4+7 v^3-7 v^2+2 v}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{14}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{13}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^9+a^9 z^{-1} -3 z^3 a^7-5 z a^7-3 a^7 z^{-1} +2 z^5 a^5+5 z^3 a^5+6 z a^5+4 a^5 z^{-1} +z^5 a^3-3 z a^3-2 a^3 z^{-1} -z^3 a-z a }[/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}-3 z^2 a^{10}+a^{10}-5 z^7 a^9+8 z^5 a^9-5 z^3 a^9+3 z a^9-a^9 z^{-1} -4 z^8 a^8+10 z^4 a^8-8 z^2 a^8+3 a^8-z^9 a^7-13 z^7 a^7+33 z^5 a^7-31 z^3 a^7+14 z a^7-3 a^7 z^{-1} -8 z^8 a^6+7 z^6 a^6+7 z^4 a^6-9 z^2 a^6+3 a^6-z^9 a^5-13 z^7 a^5+34 z^5 a^5-34 z^3 a^5+17 z a^5-4 a^5 z^{-1} -4 z^8 a^4+z^6 a^4+7 z^4 a^4-6 z^2 a^4+2 a^4-5 z^7 a^3+9 z^5 a^3-8 z^3 a^3+6 z a^3-2 a^3 z^{-1} -3 z^6 a^2+5 z^4 a^2-2 z^2 a^2-z^5 a+2 z^3 a-z a }[/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 χ 2                     1 1 0                   2   -2 -2                 5 1   4 -4               6 3     -3 -6             8 4       4 -8           6 6         0 -10         7 8           -1 -12       5 7             2 -14     2 6               -4 -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}^{2} }[/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}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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