L11n385

L11n384

L11n386

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Link Presentations

[edit Notes on L11n385's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X13,2,14,3
Gauss code	{1, 11, -5, -3}, {-9, 8, -10, 7}, {-4, -1, 2, 5, -6, 9, -8, 4, -11, -2, 3, 6, -7, 10}

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{(v-1) (w-1) \left(u w^4-1\right)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-1} - q^{-2} +3 q^{-3} - q^{-4} +3 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{10}-a^{10} z^{-2} -2 a^{10}+z^6 a^8+6 z^4 a^8+11 z^2 a^8+4 a^8 z^{-2} +10 a^8-z^8 a^6-7 z^6 a^6-17 z^4 a^6-21 z^2 a^6-5 a^6 z^{-2} -16 a^6+z^6 a^4+6 z^4 a^4+11 z^2 a^4+2 a^4 z^{-2} +8 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{13} z+a^{12} z^2+a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +2 a^{10} z^4-7 a^{10} z^2-a^{10} z^{-2} +4 a^{10}+2 a^9 z^7-11 a^9 z^5+19 a^9 z^3-17 a^9 z+5 a^9 z^{-1} +3 a^8 z^8-19 a^8 z^6+39 a^8 z^4-36 a^8 z^2-4 a^8 z^{-2} +17 a^8+a^7 z^9-3 a^7 z^7-8 a^7 z^5+30 a^7 z^3-29 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-26 a^6 z^6+54 a^6 z^4-47 a^6 z^2-5 a^6 z^{-2} +22 a^6+a^5 z^9-5 a^5 z^7+3 a^5 z^5+12 a^5 z^3-15 a^5 z+5 a^5 z^{-1} +a^4 z^8-7 a^4 z^6+17 a^4 z^4-19 a^4 z^2-2 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 1 2 χ -1                   1 1 -3                     0 -5               3 1   2 -7           1 1 2     2 -9           3 1       2 -11       2 2 1         1 -13     1 4 3           0 -15   1 1 1             1 -17   2 2               0 -19 1 1                 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=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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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