L11n429

L11n428

L11n430

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Link Presentations

[edit Notes on L11n429's Link Presentations]

Planar diagram presentation	X8192 X9,20,10,21 X14,5,15,6 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X6,15,1,16 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -7, 10, 3, -9}, {5, -1, -2, 7, 8, -4}, {4, -3, 9, -5, 6, -8, -10, 2, 11, -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{u^2 v^2 w^3-2 u^2 v^2 w^2+u^2 v^2 w-u^2 v w^3+2 u^2 v w^2+u v^2 w^2-u v^2 w-u v w^2+u v w+u w^2-u w-2 v w+v-w^2+2 w-1}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q+3-4 q^{-1} +6 q^{-2} -6 q^{-3} +7 q^{-4} -5 q^{-5} +5 q^{-6} -2 q^{-7} + q^{-8} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^2+a^8 z^{-2} +a^8-a^6 z^6-5 a^6 z^4-7 a^6 z^2-2 a^6 z^{-2} -5 a^6+a^4 z^8+6 a^4 z^6+12 a^4 z^4+11 a^4 z^2+a^4 z^{-2} +4 a^4-a^2 z^6-4 a^2 z^4-3 a^2 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^2+2 a^9 z^3+5 a^8 z^4-9 a^8 z^2-a^8 z^{-2} +6 a^8+2 a^7 z^7-5 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-18 a^6 z^6+31 a^6 z^4-29 a^6 z^2-2 a^6 z^{-2} +12 a^6+2 a^5 z^9-5 a^5 z^7-3 a^5 z^5+10 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +7 a^4 z^8-32 a^4 z^6+44 a^4 z^4-26 a^4 z^2-a^4 z^{-2} +8 a^4+2 a^3 z^9-6 a^3 z^7-2 a^3 z^5+10 a^3 z^3-3 a^3 z+3 a^2 z^8-14 a^2 z^6+18 a^2 z^4-7 a^2 z^2+a^2+a z^7-4 a z^5+4 a z^3-a z }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                   1 -1 1                 2   2 -1               2 1   -1 -3             4 2     2 -5           3 3       0 -7         4 3         1 -9       2 4           2 -11     3 3             0 -13     3               3 -15 1 2                 -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=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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