L11n338

L11n337

L11n339

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Link Presentations

[edit Notes on L11n338's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X7,14,8,15 X13,8,14,9 X22,20,13,19 X20,16,21,15 X16,22,17,21 X17,12,18,5 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) \left(t(1) t(3)^3+t(1) t(2) t(3)^3-2 t(1) t(2) t(3)^2+2 t(2) t(3)^2+2 t(1) t(2) t(3)-2 t(2) t(3)+t(2)^2+t(2)\right)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^2+3 q-5+6 q^{-1} -6 q^{-2} +6 q^{-3} -4 q^{-4} +3 q^{-5} + q^{-6} + q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 a^8 z^{-2} +a^8-2 a^6 z^2-5 a^6 z^{-2} -7 a^6+3 a^4 z^2+4 a^4 z^{-2} +6 a^4+a^2 z^6+4 a^2 z^4+5 a^2 z^2-a^2 z^{-2} +a^2-z^4-2 z^2-1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^8-8 a^8 z^6+21 a^8 z^4-24 a^8 z^2-2 a^8 z^{-2} +12 a^8+a^7 z^7-10 a^7 z^5+25 a^7 z^3-21 a^7 z+5 a^7 z^{-1} +a^6 z^8-11 a^6 z^6+35 a^6 z^4-41 a^6 z^2-5 a^6 z^{-2} +23 a^6+2 a^5 z^7-15 a^5 z^5+42 a^5 z^3-35 a^5 z+9 a^5 z^{-1} +a^4 z^8-3 a^4 z^6+9 a^4 z^4-12 a^4 z^2-4 a^4 z^{-2} +12 a^4+4 a^3 z^7-11 a^3 z^5+17 a^3 z^3-15 a^3 z+5 a^3 z^{-1} +a^2 z^8+3 a^2 z^6-12 a^2 z^4+8 a^2 z^2-a^2 z^{-2} -a^2+3 a z^7-5 a z^5+z^5 a^{-1} -2 a z^3-2 z^3 a^{-1} +a z^{-1} +z a^{-1} +3 z^6-7 z^4+3 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     2   2 1                   3 1   -2 -1                 3 2     1 -3               4 4       0 -5           1 3 2         0 -7           2 4           2 -9       1 3 3             -1 -11         4               4 -13     1                   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=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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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