L11a410

L11a409

L11a411

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Link Presentations

[edit Notes on L11a410's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X10,12,5,11 X22,18,11,17 X16,22,17,21 X8,19,9,20 X18,9,19,10 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -8, 9, -5}, {5, -2, 11, -4, 3, -7, 6, -9, 8, -3, 7, -6}

A Braid Representative	{{{braid_table}}}
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(1) t(2)^2 t(3)^4-t(2)^2 t(3)^4+t(1) t(3)^4-2 t(1) t(2) t(3)^4+t(2) t(3)^4-t(1) t(2)^2 t(3)^3+2 t(2)^2 t(3)^3-2 t(1) t(3)^3+2 t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+2 t(1) t(3)^2-2 t(1) t(2) t(3)^2+2 t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-2 t(2) t(3)+t(3)-t(2)^2+t(1)-t(1) t(2)+2 t(2)-1}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-6 q^7+9 q^6-12 q^5+13 q^4-12 q^3+12 q^2-7 q+6-2 q^{-1} + q^{-2} }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +14 z^4 a^{-4} -4 z^4 a^{-6} +z^4-17 z^2 a^{-2} +17 z^2 a^{-4} -5 z^2 a^{-6} +4 z^2-13 a^{-2} +11 a^{-4} -3 a^{-6} +5-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-11} +3 z^4 a^{-10} +6 z^5 a^{-9} -4 z^3 a^{-9} +z a^{-9} +9 z^6 a^{-8} -12 z^4 a^{-8} +3 z^2 a^{-8} +11 z^7 a^{-7} -23 z^5 a^{-7} +12 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +10 z^8 a^{-6} -26 z^6 a^{-6} +19 z^4 a^{-6} -12 z^2 a^{-6} - a^{-6} z^{-2} +5 a^{-6} +5 z^9 a^{-5} -4 z^7 a^{-5} -30 z^5 a^{-5} +43 z^3 a^{-5} -21 z a^{-5} +5 a^{-5} z^{-1} +z^{10} a^{-4} +12 z^8 a^{-4} -62 z^6 a^{-4} +87 z^4 a^{-4} -52 z^2 a^{-4} -4 a^{-4} z^{-2} +18 a^{-4} +7 z^9 a^{-3} -24 z^7 a^{-3} +9 z^5 a^{-3} +31 z^3 a^{-3} -29 z a^{-3} +9 a^{-3} z^{-1} +z^{10} a^{-2} +3 z^8 a^{-2} -33 z^6 a^{-2} +67 z^4 a^{-2} -53 z^2 a^{-2} -5 a^{-2} z^{-2} +21 a^{-2} +2 z^9 a^{-1} -9 z^7 a^{-1} +10 z^5 a^{-1} +5 z^3 a^{-1} -13 z a^{-1} +5 a^{-1} z^{-1} +z^8-6 z^6+14 z^4-16 z^2-2 z^{-2} +9 }[/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 19                       1 -1 17                     2   2 15                   4 1   -3 13                 5 2     3 11               7 4       -3 9             6 5         1 7           7 8           1 5         5 5             0 3       4 9               5 1     2 3                 -1 -1   1 5                   4 -3   1                     -1 -5 1                       1

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=5 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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