L11n407

L11n406

L11n408

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Link Presentations

[edit Notes on L11n407's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X22,12,19,11 X10,4,11,3 X5,21,6,20 X21,5,22,18 X12,20,13,19 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code	{1, -10, 5, -3}, {8, 6, -7, -4}, {-6, -1, 2, -11, 9, -5, 4, -8, 10, -9, 11, -2, 3, 7}

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{2 (t(1)-1) (t(3)-1)^2 (t(2) t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 3 q^5-5 q^4+9 q^3-9 q^2+11 q-10+8 q^{-1} -5 q^{-2} +3 q^{-3} - q^{-4} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{-6} z^{-2} + a^{-6} -z^4 a^{-4} -3 z^2 a^{-4} -2 a^{-4} z^{-2} -4 a^{-4} +z^6 a^{-2} -a^2 z^4+3 z^4 a^{-2} -2 a^2 z^2+3 z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2} +z^6+3 z^4+2 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+6 z^8 a^{-2} +9 z^8+a^3 z^7-3 a z^7+4 z^7 a^{-1} +8 z^7 a^{-3} -13 a^2 z^6-12 z^6 a^{-2} +7 z^6 a^{-4} -32 z^6-4 a^3 z^5-10 a z^5-25 z^5 a^{-1} -16 z^5 a^{-3} +3 z^5 a^{-5} +17 a^2 z^4+4 z^4 a^{-2} -12 z^4 a^{-4} +33 z^4+5 a^3 z^3+18 a z^3+20 z^3 a^{-1} +7 z^3 a^{-3} -6 a^2 z^2-3 z^2 a^{-2} +11 z^2 a^{-4} +6 z^2 a^{-6} -14 z^2-2 a^3 z-6 a z-6 z a^{-1} +2 z a^{-3} +4 z a^{-5} +a^2- a^{-2} -6 a^{-4} -4 a^{-6} +3-2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} 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 11                   3 3 9                 4 2 -2 7               5 1   4 5             4 4     0 3           7 5       2 1         5 6         1 -1       3 5           -2 -3     2 5             3 -5   1 3               -2 -7   2                 2 -9 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=-5 }[/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} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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