L11n430

L11n429

L11n431

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Link Presentations

[edit Notes on L11n430's Link Presentations]

Planar diagram presentation	X8192 X9,20,10,21 X14,5,15,6 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X6,15,1,16 X4,20,5,19 X21,3,22,2
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{(v-1) (w-1) \left(u^2 w^2-u v w+u v-u w^2+u w-v\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^3+3 q^2-5 q+8-7 q^{-1} +9 q^{-2} -6 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4+2 a^4 z^2+a^4 z^{-2} +a^4-a^2 z^6-4 a^2 z^4-6 a^2 z^2-z^2 a^{-2} -2 a^2 z^{-2} -4 a^2- a^{-2} +2 z^4+5 z^2+ z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+a^6 z^2+3 a^5 z^7-10 a^5 z^5+6 a^5 z^3+4 a^4 z^8-15 a^4 z^6+16 a^4 z^4-9 a^4 z^2-a^4 z^{-2} +3 a^4+2 a^3 z^9-4 a^3 z^7-3 a^3 z^5+5 a^3 z^3+z^3 a^{-3} -2 a^3 z-z a^{-3} +2 a^3 z^{-1} +7 a^2 z^8-29 a^2 z^6+43 a^2 z^4+3 z^4 a^{-2} -27 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+2 a^{-2} +2 a z^9-6 a z^7+z^7 a^{-1} +6 a z^5-z^5 a^{-1} +2 a z^3+4 z^3 a^{-1} -4 a z-3 z a^{-1} +2 a z^{-1} +3 z^8-13 z^6+27 z^4-21 z^2- z^{-2} +7 }[/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 7                   1 -1 5                 2   2 3               3 1   -2 1             5 2     3 -1           4 5       1 -3         5 3         2 -5       3 6           3 -7     2 3             -1 -9   1 3               2 -11   2                 -2 -13 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=-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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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