L11n277

L11n276

L11n278

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Link Presentations

[edit Notes on L11n277's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X22,17,11,18 X20,11,21,12 X16,21,17,22 X15,1,16,4 X19,10,20,5
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, {8, 2, 4, -5, -10, -9, 7, 6, -11, -8, 9, -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{u v w^4-u v w^3+u v w^2-u v w+u w-v^2 w^3+v w^3-v w^2+v w-v}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{-2} - q^{-3} +4 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +2 q^{-8} - q^{-9} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+z^4 a^8+5 z^2 a^8+4 a^8 z^{-2} +8 a^8-z^6 a^6-6 z^4 a^6-14 z^2 a^6-5 a^6 z^{-2} -15 a^6+2 z^4 a^4+8 z^2 a^4+2 a^4 z^{-2} +8 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{11}-2 z a^{11}+a^{11} z^{-1} +2 z^4 a^{10}-3 z^2 a^{10}-a^{10} z^{-2} +2 a^{10}+z^7 a^9-5 z^5 a^9+14 z^3 a^9-13 z a^9+5 a^9 z^{-1} +z^8 a^8-6 z^6 a^8+18 z^4 a^8-21 z^2 a^8-4 a^8 z^{-2} +13 a^8+2 z^7 a^7-10 z^5 a^7+26 z^3 a^7-27 z a^7+9 a^7 z^{-1} +z^8 a^6-6 z^6 a^6+19 z^4 a^6-29 z^2 a^6-5 a^6 z^{-2} +20 a^6+z^7 a^5-5 z^5 a^5+13 z^3 a^5-16 z a^5+5 a^5 z^{-1} +3 z^4 a^4-11 z^2 a^4-2 a^4 z^{-2} +10 a^4 }[/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 χ -3               2 2 -5             1 2 1 -7           3     3 -9         1 2     1 -11       3 2       1 -13       1         1 -15   2 3           -1 -17   1             1 -19 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{ 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}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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