L11n296

L11n295

L11n297

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Link Presentations

[edit Notes on L11n296's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X15,20,16,21 X14,8,15,7 X10,22,5,21 X18,11,19,12 X9,17,10,16 X22,17,11,18 X19,9,20,8 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, {6, -2, 11, -4, -3, 7, 8, -6, -9, 3, 5, -8}

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 v^2 w+u v w^4-u v w^3+u v w^2-2 u v w+u v-u w^4+u w^3-v^2 w+v^2-v w^4+2 v w^3-v w^2+v w-v-w^3}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-2 q^5+4 q^4-4 q^3+7 q^2-5 q+6-4 q^{-1} +2 q^{-2} - q^{-3} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-4} +3 z^2 a^{-4} +2 a^{-4} z^{-2} +3 a^{-4} -z^6 a^{-2} -5 z^4 a^{-2} -a^2 z^2-10 z^2 a^{-2} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^2-10 a^{-2} +2 z^4+7 z^2+4 z^{-2} +9 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+a z^7-z^7 a^{-1} +2 z^7 a^{-5} -25 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -10 z^6-4 a z^5-10 z^5 a^{-1} -13 z^5 a^{-3} -7 z^5 a^{-5} +2 a^2 z^4+48 z^4 a^{-2} +23 z^4 a^{-4} -4 z^4 a^{-6} +23 z^4+a^3 z^3+12 a z^3+30 z^3 a^{-1} +23 z^3 a^{-3} +4 z^3 a^{-5} -3 a^2 z^2-44 z^2 a^{-2} -22 z^2 a^{-4} +3 z^2 a^{-6} -22 z^2-2 a^3 z-13 a z-27 z a^{-1} -16 z a^{-3} +2 a^2+20 a^{-2} +10 a^{-4} +13+a^3 z^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +5 a^{-3} z^{-1} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^{-4} z^{-2} -4 z^{-2} }[/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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 13                   1 1 11                 2 1 -1 9               2     2 7             2 2     0 5           5 2       3 3         1 3         2 1       5 4           1 -1     1 3             2 -3   1 3               -2 -5   1                 1 -7 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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