L11n304

L11n303

L11n305

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Link Presentations

[edit Notes on L11n304's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,13,21,14 X22,19,11,20 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X14,21,15,22 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}

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 v^2 w^3+u v^2 w^2+u v w^3-2 u v w^2+u v w+2 u w^2-2 u w+u-v^2 w^4+2 v^2 w^3-2 v^2 w^2-v w^3+2 v w^2-v w-w^2+w}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-7} +2 q^{-6} -4 q^{-5} +6 q^{-4} -6 q^{-3} +2 q^2+8 q^{-2} -3 q-6 q^{-1} +6 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6-a^6 z^{-2} -3 a^6+3 z^4 a^4+12 z^2 a^4+4 a^4 z^{-2} +13 a^4-2 z^6 a^2-11 z^4 a^2-21 z^2 a^2-5 a^2 z^{-2} -17 a^2+2 z^4+7 z^2+2 z^{-2} +7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-5 a^7 z^5+8 a^7 z^3-5 a^7 z+a^7 z^{-1} +2 a^6 z^8-9 a^6 z^6+12 a^6 z^4-7 a^6 z^2-a^6 z^{-2} +4 a^6+a^5 z^9+a^5 z^7-20 a^5 z^5+35 a^5 z^3-21 a^5 z+5 a^5 z^{-1} +6 a^4 z^8-26 a^4 z^6+38 a^4 z^4-32 a^4 z^2-4 a^4 z^{-2} +17 a^4+a^3 z^9+4 a^3 z^7-28 a^3 z^5+44 a^3 z^3-33 a^3 z+9 a^3 z^{-1} +4 a^2 z^8-15 a^2 z^6+26 a^2 z^4-32 a^2 z^2+3 z^2 a^{-2} -5 a^2 z^{-2} +20 a^2-2 a^{-2} +4 a z^7-12 a z^5+z^5 a^{-1} +18 a z^3+z^3 a^{-1} -16 a z+z a^{-1} +5 a z^{-1} +2 z^6-4 z^2-2 z^{-2} +6 }[/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 χ 5                   2 2 3                 3 2 -1 1               3     3 -1             4 4     0 -3           4 2       2 -5         2 4         2 -7       4 4           0 -9     1 3             2 -11   1 3               -2 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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