L11n445

L11n444

L11n446

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Link Presentations

[edit Notes on L11n445's Link Presentations]

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

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) (t(4)-1) \left(t(4)^2+t(1) t(4)-2 t(1) t(2) t(4)+t(2) t(4)-2 t(4)+t(1) t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -10 q^{9/2}+9 q^{7/2}-12 q^{5/2}+9 q^{3/2}-\frac{2}{q^{3/2}}+q^{15/2}-3 q^{13/2}+6 q^{11/2}-9 \sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} + a^{-7} z^{-1} -3 z^3 a^{-5} - a^{-5} z^{-3} -6 z a^{-5} -5 a^{-5} z^{-1} +2 z^5 a^{-3} +7 z^3 a^{-3} +3 a^{-3} z^{-3} +12 z a^{-3} +10 a^{-3} z^{-1} -4 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +2 a z-9 z a^{-1} +3 a z^{-1} -9 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-3} -z^9 a^{-5} -4 z^8 a^{-2} -7 z^8 a^{-4} -3 z^8 a^{-6} -4 z^7 a^{-1} -10 z^7 a^{-3} -9 z^7 a^{-5} -3 z^7 a^{-7} +8 z^6 a^{-2} +13 z^6 a^{-4} +3 z^6 a^{-6} -z^6 a^{-8} -z^6+14 z^5 a^{-1} +43 z^5 a^{-3} +38 z^5 a^{-5} +9 z^5 a^{-7} +9 z^4 a^{-4} +12 z^4 a^{-6} +3 z^4 a^{-8} -3 a z^3-33 z^3 a^{-1} -66 z^3 a^{-3} -45 z^3 a^{-5} -9 z^3 a^{-7} -24 z^2 a^{-2} -30 z^2 a^{-4} -14 z^2 a^{-6} -3 z^2 a^{-8} -5 z^2+7 a z+30 z a^{-1} +48 z a^{-3} +31 z a^{-5} +6 z a^{-7} +21 a^{-2} +18 a^{-4} +6 a^{-6} + a^{-8} +9-5 a z^{-1} -14 a^{-1} z^{-1} -18 a^{-3} z^{-1} -11 a^{-5} z^{-1} -2 a^{-7} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} 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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 2   2 12               4 1   -3 10             6 2     4 8           5 6       1 6         7 4         3 4     1 3 5           3 2     7 7             0 0   2 8               6 -2   1                 -1 -4 2                   2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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