L11n433

L11n432

L11n434

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Link Presentations

[edit Notes on L11n433's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X5,17,6,16 X11,19,12,18 X17,11,18,10 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{(u v w+1) \left(u v w^2-u w^2+u-v w^2+v-1\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{10}-q^9+q^8-q^7+q^6+q^5+2 q^3-q^2+q }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -16 z^4 a^{-6} +6 z^4 a^{-8} +10 z^2 a^{-4} -17 z^2 a^{-6} +8 z^2 a^{-8} -z^2 a^{-10} +6 a^{-4} -9 a^{-6} +3 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-12} -3 z^2 a^{-12} +z^5 a^{-11} -3 z^3 a^{-11} +z^6 a^{-10} -5 z^4 a^{-10} +5 z^2 a^{-10} -2 a^{-10} +z^7 a^{-9} -6 z^5 a^{-9} +7 z^3 a^{-9} +2 z^8 a^{-8} -13 z^6 a^{-8} +22 z^4 a^{-8} -11 z^2 a^{-8} - a^{-8} z^{-2} +3 a^{-8} +z^9 a^{-7} -5 z^7 a^{-7} +z^5 a^{-7} +13 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} +3 z^8 a^{-6} -21 z^6 a^{-6} +44 z^4 a^{-6} -35 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} +z^9 a^{-5} -6 z^7 a^{-5} +8 z^5 a^{-5} +3 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +16 z^4 a^{-4} -16 z^2 a^{-4} - a^{-4} z^{-2} +7 a^{-4} }[/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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   1 1 0 17               1 1     0 15             1 2 1     0 13           1 2 1       0 11         1 1 2         2 9       1 3 1           1 7     1 1 2             2 5   1 2                 1 3                       0 1 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=5 }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/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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