L11n451

L11n450

L11n452

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Link Presentations

[edit Notes on L11n451's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X20,12,21,11 X22,13,19,14 X18,22,9,21 X12,17,13,18 X15,8,16,5 X7,14,8,15 X16,19,17,20 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -8, 7}, {9, -3, 5, -4}, {-11, 2, 3, -6, 4, 8, -7, -9, 6, -5}

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{(w-1) (x-1)^2 (u x-v)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+q^{3/2}-\sqrt{q}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+a^5 z^{-3} +7 a^5 z+5 a^5 z^{-1} -2 a^3 z^5-9 a^3 z^3-3 a^3 z^{-3} -15 a^3 z-10 a^3 z^{-1} +a z^5+7 a z^3+3 a z^{-3} -z^3 a^{-1} - a^{-1} z^{-3} +12 a z+9 a z^{-1} -3 z a^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8-3 z^7 a^7+10 z^5 a^7-9 z^3 a^7+5 z a^7-2 a^7 z^{-1} -2 z^8 a^6+z^6 a^6+15 z^4 a^6-15 z^2 a^6+6 a^6-9 z^7 a^5+35 z^5 a^5-43 z^3 a^5+31 z a^5-11 a^5 z^{-1} +a^5 z^{-3} -2 z^8 a^4+z^6 a^4+19 z^4 a^4-33 z^2 a^4-3 a^4 z^{-2} +18 a^4-7 z^7 a^3+39 z^5 a^3-73 z^3 a^3+54 z a^3-18 a^3 z^{-1} +3 a^3 z^{-3} -z^8 a^2+5 z^6 a^2+2 z^4 a^2-27 z^2 a^2-6 a^2 z^{-2} +21 a^2-2 z^7 a+20 z^5 a-49 z^3 a+38 z a-14 a z^{-1} +3 a z^{-3} -z^8+6 z^6-5 z^4-6 z^2-3 z^{-2} +9-z^7 a^{-1} +6 z^5 a^{-1} -10 z^3 a^{-1} +10 z a^{-1} -5 a^{-1} z^{-1} + a^{-1} z^{-3} }[/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 6                       1 1 4                         0 2                   1 1   0 0               6 1       5 -2             4 6 1       1 -4           4 2 2         4 -6         2 4             2 -8       5 4 1             2 -10     2 4                 2 -12   1 3                   -2 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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