L11n59

L11n58

L11n60

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Link Presentations

[edit Notes on L11n59's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X11,19,12,18 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}

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(2)^5-t(1) t(2)^4+t(2)^4+4 t(1) t(2)^3-4 t(2)^3-4 t(1) t(2)^2+4 t(2)^2+t(1) t(2)-t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -6 q^{9/2}+7 q^{7/2}-6 q^{5/2}-\frac{1}{q^{5/2}}+5 q^{3/2}+q^{15/2}-3 q^{13/2}+5 q^{11/2}-4 \sqrt{q} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} +z^5 a^{-3} +a z^3-7 z^3 a^{-1} +3 z^3 a^{-3} -2 z^3 a^{-5} +4 a z-10 z a^{-1} +5 z a^{-3} -2 z a^{-5} +z a^{-7} +3 a z^{-1} -5 a^{-1} z^{-1} +2 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-8} -3 z^4 a^{-8} +2 z^2 a^{-8} +3 z^7 a^{-7} -10 z^5 a^{-7} +7 z^3 a^{-7} -z a^{-7} +3 z^8 a^{-6} -9 z^6 a^{-6} +5 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +z^9 a^{-5} +z^7 a^{-5} -11 z^5 a^{-5} +8 z^3 a^{-5} -z a^{-5} +4 z^8 a^{-4} -15 z^6 a^{-4} +19 z^4 a^{-4} -10 z^2 a^{-4} +z^9 a^{-3} -3 z^7 a^{-3} +2 z^5 a^{-3} +3 z^3 a^{-3} -3 z a^{-3} +2 a^{-3} z^{-1} +z^8 a^{-2} -5 z^6 a^{-2} +7 z^4 a^{-2} +5 z^2 a^{-2} -5 a^{-2} +a z^7-7 a z^5-4 z^5 a^{-1} +14 a z^3+16 z^3 a^{-1} -11 a z-14 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} -4 z^4+10 z^2-5 }[/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 6 7 χ 16                       1 -1 14                     2   2 12                   3 1   -2 10                 3 2     1 8               4 3       -1 6           1 3 3         -1 4           3 4           1 2       1 3 3             -1 0         4               4 -2     1 1                 0 -4 1                       1 -6 1                       1

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