L11n369

L11n368

L11n370

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Link Presentations

[edit Notes on L11n369's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X20,16,21,15 X18,11,19,12 X12,17,13,18 X22,20,17,19 X16,22,5,21 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -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-1) (v-1) (w-1)^3}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^2+4 q-7+10 q^{-1} -10 q^{-2} +12 q^{-3} -9 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^6+a^6 z^{-2} +a^6-2 z^4 a^4-4 z^2 a^4-2 a^4 z^{-2} -4 a^4+z^6 a^2+3 z^4 a^2+4 z^2 a^2+a^2 z^{-2} +3 a^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^4-2 a^8 z^2+a^8+3 a^7 z^5-3 a^7 z^3+a^6 z^8-a^6 z^6+6 a^6 z^4-4 a^6 z^2+a^6 z^{-2} -a^6+a^5 z^9+3 a^5 z^5-3 a^5 z^3+4 a^5 z-2 a^5 z^{-1} +5 a^4 z^8-8 a^4 z^6+6 a^4 z^4+a^4 z^2+2 a^4 z^{-2} -4 a^4+a^3 z^9+6 a^3 z^7-12 a^3 z^5+3 a^3 z^3+4 a^3 z-2 a^3 z^{-1} +4 a^2 z^8-3 a^2 z^6-6 a^2 z^4+5 a^2 z^2+a^2 z^{-2} -3 a^2+6 a z^7-11 a z^5+z^5 a^{-1} +2 a z^3-z^3 a^{-1} +4 z^6-7 z^4+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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 3   3 1               4 1   -3 -1             6 3     3 -3           6 6       0 -5         6 4         2 -7       3 6           3 -9     4 6             -2 -11   1 5               4 -13   2                 -2 -15 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=-1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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