L11n431

L11n430

L11n432

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Link Presentations

[edit Notes on L11n431's Link Presentations]

Planar diagram presentation	X8192 X20,10,21,9 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X10,4,11,3 X18,11,19,12 X15,1,16,6 X4,20,5,19 X2,21,3,22
Gauss code	{1, -11, 7, -10, -3, 9}, {5, -1, 2, -7, 8, -4}, {4, 3, -9, -5, 6, -8, 10, -2, 11, -6}

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{(w-1) \left(-u^2 v^2 w+u^2 v^2+u^2 v w-u^2 v+u v^2 w-u v^2+u v w^2-3 u v w+u v-u w^2+u w-v w^2+v w+w^2-w\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9-4 q^8+7 q^7-9 q^6+12 q^5-11 q^4+11 q^3-7 q^2+5 q-1 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-8} -2 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} z^{-2} +z^6 a^{-4} +3 z^4 a^{-4} +3 z^2 a^{-4} -2 a^{-4} z^{-2} -2 a^{-4} -z^4 a^{-2} + a^{-2} z^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-10} -2 z^4 a^{-10} +4 z^7 a^{-9} -11 z^5 a^{-9} +5 z^3 a^{-9} +6 z^8 a^{-8} -19 z^6 a^{-8} +15 z^4 a^{-8} -4 z^2 a^{-8} +3 z^9 a^{-7} -3 z^7 a^{-7} -10 z^5 a^{-7} +8 z^3 a^{-7} +11 z^8 a^{-6} -36 z^6 a^{-6} +37 z^4 a^{-6} -11 z^2 a^{-6} + a^{-6} z^{-2} -2 a^{-6} +3 z^9 a^{-5} -5 z^7 a^{-5} +z^5 a^{-5} +2 z^3 a^{-5} +2 z a^{-5} -2 a^{-5} z^{-1} +5 z^8 a^{-4} -16 z^6 a^{-4} +25 z^4 a^{-4} -10 z^2 a^{-4} +2 a^{-4} z^{-2} -3 a^{-4} +2 z^7 a^{-3} +2 z a^{-3} -2 a^{-3} z^{-1} +5 z^4 a^{-2} -3 z^2 a^{-2} + a^{-2} z^{-2} -2 a^{-2} +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   \ -1 0 1 2 3 4 5 6 7 8 χ 19                   1 1 17                 3   -3 15               4 1   3 13             5 3     -2 11           7 4       3 9         5 6         1 7       6 6           0 5     3 7             4 3   2 4               -2 1   4                 4 -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=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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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