L11n434

L11n433

L11n435

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Link Presentations

[edit Notes on L11n434's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X15,20,16,21 X5,17,6,16 X18,11,19,12 X10,17,11,18 X19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, -7, 8, 10, -9, -11, 7, 6, -5}

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{2 u^2 v w-u^2 v-u^2 w+u^2+u v^2 w^3-u v^2 w^2+u v^2 w-u v^2-2 u v w^3+3 u v w^2-3 u v w+2 u v+u w^3-u w^2+u w-u-v^2 w^3+v^2 w^2+v w^3-2 v w^2}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+6 q^4-7 q^3- q^{-3} +10 q^2+4 q^{-2} -9 q-6 q^{-1} +9 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +2 z^4-a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +4 z^2-5 a^{-2} +2 a^{-4} +3-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-6} -3 z^4 a^{-6} +2 z^2 a^{-6} +3 z^7 a^{-5} -9 z^5 a^{-5} +5 z^3 a^{-5} +4 z^8 a^{-4} -13 z^6 a^{-4} +13 z^4 a^{-4} -11 z^2 a^{-4} - a^{-4} z^{-2} +6 a^{-4} +2 z^9 a^{-3} -2 z^7 a^{-3} -7 z^5 a^{-3} +a^3 z^3+8 z^3 a^{-3} -a^3 z-6 z a^{-3} +2 a^{-3} z^{-1} +8 z^8 a^{-2} -30 z^6 a^{-2} +4 a^2 z^4+44 z^4 a^{-2} -4 a^2 z^2-33 z^2 a^{-2} -2 a^{-2} z^{-2} +a^2+12 a^{-2} +2 z^9 a^{-1} +2 a z^7-3 z^7 a^{-1} -4 a z^5-2 z^5 a^{-1} +8 a z^3+10 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-16 z^6+32 z^4-24 z^2- z^{-2} +8 }[/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   -2 9               4 1   3 7             4 3     -1 5           6 3       3 3         4 5         1 1       5 5           0 -1     2 5             3 -3   2 4               -2 -5   3                 3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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

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