L11n363

L11n362

L11n364

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Link Presentations

[edit Notes on L11n363's Link Presentations]

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

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{(t(1)-1) (t(3)-1)^2}{\sqrt{t(1)} t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-8} +2 q^{-7} -3 q^{-6} +4 q^{-5} -3 q^{-4} +4 q^{-3} +q^2- q^{-2} -q+ q^{-1} +1 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^6-3 z^2 a^6-3 a^6+z^6 a^4+6 z^4 a^4+13 z^2 a^4+a^4 z^{-2} +9 a^4-z^6 a^2-7 z^4 a^2-14 z^2 a^2-2 a^2 z^{-2} -10 a^2+z^4+4 z^2+ z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+2 z^7 a^7-6 z^5 a^7+3 z^3 a^7+z^8 a^6-3 z^6 a^6+3 z^4 a^6-3 z^2 a^6+2 a^6+2 z^7 a^5-9 z^5 a^5+14 z^3 a^5-6 z a^5+2 z^8 a^4-15 z^6 a^4+37 z^4 a^4-34 z^2 a^4-a^4 z^{-2} +13 a^4+z^9 a^3-7 z^7 a^3+11 z^5 a^3+3 z^3 a^3-9 z a^3+2 a^3 z^{-1} +2 z^8 a^2-17 z^6 a^2+43 z^4 a^2-41 z^2 a^2-2 a^2 z^{-2} +15 a^2+z^9 a-7 z^7 a+13 z^5 a-5 z^3 a-4 z a+2 a z^{-1} +z^8-7 z^6+15 z^4-13 z^2- z^{-2} +6 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                       1 1 3                         0 1                   1 1   0 -1               3 1       2 -3             1 2 1       0 -5           4 2 1         3 -7         2 3             1 -9       2 2 1             1 -11     1 2                 1 -13   1 2                   -1 -15   1                     1 -17 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/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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