L11n353

L11n352

L11n354

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Link Presentations

[edit Notes on L11n353's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,16,6,17 X8493 X9,21,10,20 X19,11,20,10 X17,22,18,15 X21,18,22,19 X15,14,16,5 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-10, 4, -8, 9, -7, 6, -9, 8}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10}

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-1) (v-1) (w-1)^3}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{-6} +5 q^{-5} -8 q^{-4} +q^3+11 q^{-3} -2 q^2-10 q^{-2} +6 q+11 q^{-1} -8 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^6-a^4 z^4+a^4+a^2 z^6+3 a^2 z^4+4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^2+2 a^{-2} -2 z^4-5 z^2-2 z^{-2} -5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+5 a^2 z^8+2 z^8+3 a^5 z^7+5 a^3 z^7+4 a z^7+2 z^7 a^{-1} +a^6 z^6-2 a^4 z^6-4 a^2 z^6+z^6 a^{-2} -2 a^5 z^5-8 a^3 z^5-11 a z^5-5 z^5 a^{-1} +4 a^6 z^4+3 a^4 z^4-11 a^2 z^4-4 z^4 a^{-2} -14 z^4+3 a^7 z^3+5 a^5 z^3-2 a^3 z^3-2 a z^3+2 z^3 a^{-1} -3 a^6 z^2-5 a^4 z^2+11 a^2 z^2+6 z^2 a^{-2} +19 z^2-2 a^7 z-3 a^5 z+3 a^3 z+7 a z+3 z a^{-1} +2 a^6+2 a^4-6 a^2-4 a^{-2} -9-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 1   -1 3               5 1   4 1             3 1     -2 -1           8 5       3 -3         6 7         1 -5       5 4           1 -7     3 6             3 -9   2 5               -3 -11   3                 3 -13 2                   -2

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=-1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/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 }[/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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