L11n317

L11n316

L11n318

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Link Presentations

[edit Notes on L11n317's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,7,17,8 X9,20,10,21 X11,18,12,19 X19,22,20,11 X8,15,9,16 X21,10,22,5 X14,18,15,17 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, -4, 8}, {-5, -2, 11, -9, 7, -3, 9, 5, -6, 4, -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{u v^2 w^2-2 u v^2 w+u v^2-3 u v w^2+5 u v w-2 u v+u w^2-3 u w+u-v^2 w^2+3 v^2 w-v^2+2 v w^2-5 v w+3 v-w^2+2 w-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^2+4 q-7+11 q^{-1} -12 q^{-2} +14 q^{-3} -11 q^{-4} +9 q^{-5} -5 q^{-6} +2 q^{-7} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 a^6 z^2+a^6 z^{-2} +2 a^6-3 a^4 z^4-7 a^4 z^2-2 a^4 z^{-2} -6 a^4+a^2 z^6+3 a^2 z^4+5 a^2 z^2+a^2 z^{-2} +4 a^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^8 z^4-4 a^8 z^2+a^8+a^7 z^7+3 a^7 z^5-4 a^7 z^3+2 a^6 z^8+a^6 z^6-3 a^6 z^4+4 a^6 z^2+a^6 z^{-2} -3 a^6+a^5 z^9+5 a^5 z^7-5 a^5 z^5-4 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +6 a^4 z^8-2 a^4 z^6-13 a^4 z^4+16 a^4 z^2+2 a^4 z^{-2} -8 a^4+a^3 z^9+10 a^3 z^7-19 a^3 z^5+4 a^3 z^3+6 a^3 z-2 a^3 z^{-1} +4 a^2 z^8+a^2 z^6-14 a^2 z^4+11 a^2 z^2+a^2 z^{-2} -5 a^2+6 a z^7-10 a z^5+z^5 a^{-1} +3 a z^3-z^3 a^{-1} +4 z^6-7 z^4+3 z^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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 3   3 1               4 1   -3 -1             7 3     4 -3           7 6       -1 -5         7 5         2 -7       5 8           3 -9     4 6             -2 -11   1 5               4 -13 1 4                 -3 -15 2                   2

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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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