L11n372

L11n371

L11n373

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Link Presentations

[edit Notes on L11n372's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,19,8,18 X15,17,16,22 X9,20,10,21 X13,9,14,8 X17,15,18,14 X21,5,22,16 X19,12,20,13 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {-7, 3, -9, 5, -8, 4}, {10, -1, -3, 6, -5, 2, -11, 9, -6, 7, -4, 8}

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(3)-1) \left(-t(1) t(3)^3+t(1) t(2) t(3)^3+t(1) t(2)^2 t(3)^2+t(1) t(3)^2-2 t(1) t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)+t(2)^2 t(3)-2 t(2) t(3)+t(3)-t(2)^2+t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^4+3 q^3-5 q^2+8 q-8+10 q^{-1} -8 q^{-2} +7 q^{-3} -4 q^{-4} +2 q^{-5} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6-a^4 z^4-3 a^4 z^2+a^4 z^{-2} -a^4+a^2 z^6+3 a^2 z^4-z^4 a^{-2} +a^2 z^2-2 a^2 z^{-2} -2 z^2 a^{-2} -2 a^2+z^6+3 z^4+2 z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 a^6 z^2-a^6+a^5 z^5+3 a^5 z^3-a^5 z+3 a^4 z^6-a^4 z^4-3 a^4 z^2-a^4 z^{-2} +4 a^4+6 a^3 z^7+z^7 a^{-3} -16 a^3 z^5-4 z^5 a^{-3} +17 a^3 z^3+5 z^3 a^{-3} -10 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +6 a^2 z^8+3 z^8 a^{-2} -20 a^2 z^6-13 z^6 a^{-2} +27 a^2 z^4+17 z^4 a^{-2} -26 a^2 z^2-8 z^2 a^{-2} -2 a^2 z^{-2} +12 a^2+2 a^{-2} +2 a z^9+2 z^9 a^{-1} +2 a z^7-3 z^7 a^{-1} -25 a z^5-12 z^5 a^{-1} +30 a z^3+21 z^3 a^{-1} -14 a z-7 z a^{-1} +2 a z^{-1} +9 z^8-36 z^6+45 z^4-28 z^2- z^{-2} +10 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 9                   1 -1 7                 2   2 5               3 1   -2 3             5 2     3 1           4 4       0 -1         6 4         2 -3       3 5           2 -5     4 5             -1 -7   1 4               3 -9 1 3                 -2 -11 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\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}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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