L11n339

L11n338

L11n340

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Link Presentations

[edit Notes on L11n339's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X7,14,8,15 X13,8,14,9 X19,22,20,13 X15,20,16,21 X21,16,22,17 X12,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, -7, 8, 9, -3, -6, 7, -8, 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{(t(2)-1) (t(3)-1) \left(t(3) t(2)^2+t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-t(2)-t(1) t(3)\right)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -3 q^{-3} +4 q^{-4} -2 q^{-5} +3 q^{-6} - q^{-8} + q^{-9} - q^{-10} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+2 z^2 a^8+4 a^8 z^{-2} +5 a^8-2 z^2 a^6-5 a^6 z^{-2} -7 a^6-z^6 a^4-4 z^4 a^4-3 z^2 a^4+2 a^4 z^{-2} +a^4+z^4 a^2+3 z^2 a^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-6 z a^{11}+a^{11} z^{-1} +z^8 a^{10}-6 z^6 a^{10}+9 z^4 a^{10}-6 z^2 a^{10}-a^{10} z^{-2} +4 a^{10}+2 z^7 a^9-16 z^5 a^9+33 z^3 a^9-22 z a^9+5 a^9 z^{-1} +z^8 a^8-9 z^6 a^8+22 z^4 a^8-22 z^2 a^8-4 a^8 z^{-2} +15 a^8+2 z^7 a^7-15 z^5 a^7+34 z^3 a^7-30 z a^7+9 a^7 z^{-1} +z^8 a^6-5 z^6 a^6+11 z^4 a^6-19 z^2 a^6-5 a^6 z^{-2} +16 a^6+3 z^7 a^5-12 z^5 a^5+15 z^3 a^5-13 z a^5+5 a^5 z^{-1} +z^8 a^4-z^6 a^4-6 z^4 a^4+2 z^2 a^4-2 a^4 z^{-2} +4 a^4+2 z^7 a^3-7 z^5 a^3+4 z^3 a^3+z a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     1   -1 -3                   3 1   2 -5                 2 3     1 -7             1 3 1       1 -9           1 1 2         2 -11         1 5 3           1 -13       1   2             3 -15       1 2               -1 -17   1 1                   0 -19                         0 -21 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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} }[/math] [math]\displaystyle{ {\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}^{5} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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]

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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