L11a433

L11a432

L11a434

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Link Presentations

[edit Notes on L11a433's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,14,19,13 X22,9,13,10 X20,11,21,12 X12,19,5,20 X10,21,11,22 X4,17,1,18
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -3, 7, -10, 8, -9}, {6, -2, 4, -5, 11, -6, 9, -8, 10, -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{(t(1)-1) (t(2)-1) (t(3)-1) \left(t(3) t(2)^2-t(2)^2+t(3)^2 t(2)-t(3) t(2)+t(2)-t(3)^2+t(3)\right)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-7} +4 q^{-6} -6 q^{-5} +q^4+11 q^{-4} -4 q^3-14 q^{-3} +8 q^2+18 q^{-2} -12 q-17 q^{-1} +16 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6+a^6 z^{-2} +2 z^4 a^4+3 z^2 a^4-2 a^4 z^{-2} -a^4-z^6 a^2-2 z^4 a^2-2 z^2 a^2+a^2 z^{-2} -z^6-2 z^4-z^2+1+z^4 a^{-2} +z^2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-3 a^7 z^5+a^7 z^3+4 a^6 z^8-16 a^6 z^6+17 a^6 z^4-3 a^6 z^2+a^6 z^{-2} -3 a^6+5 a^5 z^9-18 a^5 z^7+17 a^5 z^5-5 a^5 z^3+4 a^5 z-2 a^5 z^{-1} +2 a^4 z^{10}+4 a^4 z^8-35 a^4 z^6+45 a^4 z^4+z^4 a^{-4} -13 a^4 z^2+2 a^4 z^{-2} -4 a^4+11 a^3 z^9-34 a^3 z^7+30 a^3 z^5+4 z^5 a^{-3} -9 a^3 z^3-2 z^3 a^{-3} +4 a^3 z-2 a^3 z^{-1} +2 a^2 z^{10}+9 a^2 z^8-38 a^2 z^6+8 z^6 a^{-2} +39 a^2 z^4-8 z^4 a^{-2} -14 a^2 z^2+2 z^2 a^{-2} +a^2 z^{-2} -a^2+6 a z^9-5 a z^7+10 z^7 a^{-1} -6 a z^5-12 z^5 a^{-1} +2 a z^3+3 z^3 a^{-1} +9 z^8-11 z^6+2 z^4-2 z^2+1 }[/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 9                       1 1 7                     3   -3 5                   5 1   4 3                 7 3     -4 1               9 5       4 -1             10 9         -1 -3           8 7           1 -5         6 10             4 -7       5 8               -3 -9     3 8                 5 -11   1 3                   -2 -13   3                     3 -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=-1 }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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