L11a478

L11a477

L11a479

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Link Presentations

[edit Notes on L11a478's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X16,8,5,7 X18,9,19,10 X22,15,17,16 X14,19,15,20 X20,13,21,14 X12,21,13,22 X8,17,9,18 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {9, -4, 6, -7, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -8, 7, -6, 5, -3}

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-1) (w-1) \left(v^2 w^2-2 v^2 w+v w^3-3 v w^2+3 v w-v+2 w^2-w\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-9} -2 q^{-8} +6 q^{-7} -10 q^{-6} +15 q^{-5} -17 q^{-4} +19 q^{-3} -q^2-16 q^{-2} +4 q+13 q^{-1} -8 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^8+a^8 z^{-2} +2 a^8-2 z^4 a^6-5 z^2 a^6-2 a^6 z^{-2} -6 a^6+z^6 a^4+2 z^4 a^4+3 z^2 a^4+a^4 z^{-2} +3 a^4+z^6 a^2+2 z^4 a^2+2 z^2 a^2+a^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+5 z^2 a^{10}-2 a^{10}+2 z^7 a^9-5 z^5 a^9+2 z^3 a^9+z a^9+3 z^8 a^8-6 z^6 a^8+3 z^4 a^8-3 z^2 a^8-a^8 z^{-2} +3 a^8+3 z^9 a^7-4 z^7 a^7+4 z^3 a^7-7 z a^7+2 a^7 z^{-1} +z^{10} a^6+8 z^8 a^6-27 z^6 a^6+37 z^4 a^6-27 z^2 a^6-2 a^6 z^{-2} +11 a^6+7 z^9 a^5-9 z^7 a^5-2 z^5 a^5+12 z^3 a^5-9 z a^5+2 a^5 z^{-1} +z^{10} a^4+12 z^8 a^4-32 z^6 a^4+36 z^4 a^4-21 z^2 a^4-a^4 z^{-2} +7 a^4+4 z^9 a^3+4 z^7 a^3-19 z^5 a^3+15 z^3 a^3-2 z a^3+7 z^8 a^2-8 z^6 a^2+7 z^7 a-11 z^5 a+4 z^3 a-z a+4 z^6-6 z^4+2 z^2+z^5 a^{-1} -z^3 a^{-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 5                       1 -1 3                     3   3 1                   5 1   -4 -1                 8 3     5 -3               10 7       -3 -5             9 6         3 -7           8 10           2 -9         7 9             -2 -11       3 8               5 -13     3 7                 -4 -15   1 5                   4 -17   1                     -1 -19 1                       1

Integral Khovanov Homology (db, data source)

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