L11n438

L11n437

L11n439

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Link Presentations

[edit Notes on L11n438's Link Presentations]

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

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(4)^3-t(1) t(2) t(4)^3+t(3) t(4)^3-t(4)^3-2 t(1) t(4)^2+3 t(1) t(2) t(4)^2-t(2) t(4)^2+t(1) t(3) t(4)^2-2 t(1) t(2) t(3) t(4)^2+2 t(2) t(3) t(4)^2-3 t(3) t(4)^2+2 t(4)^2+2 t(1) t(4)-3 t(1) t(2) t(4)+t(2) t(4)-t(1) t(3) t(4)+2 t(1) t(2) t(3) t(4)-2 t(2) t(3) t(4)+3 t(3) t(4)-2 t(4)+t(1) t(2)-t(1) t(2) t(3)+t(2) t(3)-t(3)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 11 q^{9/2}-13 q^{7/2}+6 q^{5/2}-4 q^{3/2}+q^{21/2}-2 q^{19/2}+6 q^{17/2}-10 q^{15/2}+12 q^{13/2}-15 q^{11/2} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -3 z^5 a^{-5} +4 z^3 a^{-3} -12 z^3 a^{-5} +6 z^3 a^{-7} +9 z a^{-3} -22 z a^{-5} +17 z a^{-7} -4 z a^{-9} +7 a^{-3} z^{-1} -20 a^{-5} z^{-1} +20 a^{-7} z^{-1} -8 a^{-9} z^{-1} + a^{-11} z^{-1} +2 a^{-3} z^{-3} -7 a^{-5} z^{-3} +9 a^{-7} z^{-3} -5 a^{-9} z^{-3} + a^{-11} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-12} -4 z^4 a^{-12} +6 z^2 a^{-12} + a^{-12} z^{-2} -4 a^{-12} +2 z^7 a^{-11} -5 z^5 a^{-11} +4 z^3 a^{-11} - a^{-11} z^{-3} -2 z a^{-11} +2 a^{-11} z^{-1} +2 z^8 a^{-10} +2 z^6 a^{-10} -21 z^4 a^{-10} +34 z^2 a^{-10} +7 a^{-10} z^{-2} -24 a^{-10} +z^9 a^{-9} +8 z^7 a^{-9} -25 z^5 a^{-9} +22 z^3 a^{-9} -5 a^{-9} z^{-3} -16 z a^{-9} +12 a^{-9} z^{-1} +7 z^8 a^{-8} -46 z^4 a^{-8} +75 z^2 a^{-8} +18 a^{-8} z^{-2} -58 a^{-8} +z^9 a^{-7} +16 z^7 a^{-7} -46 z^5 a^{-7} +50 z^3 a^{-7} -9 a^{-7} z^{-3} -37 z a^{-7} +24 a^{-7} z^{-1} +5 z^8 a^{-6} +5 z^6 a^{-6} -41 z^4 a^{-6} +73 z^2 a^{-6} +19 a^{-6} z^{-2} -60 a^{-6} +10 z^7 a^{-5} -26 z^5 a^{-5} +42 z^3 a^{-5} -7 a^{-5} z^{-3} -39 z a^{-5} +23 a^{-5} z^{-1} +6 z^6 a^{-4} -12 z^4 a^{-4} +26 z^2 a^{-4} +7 a^{-4} z^{-2} -23 a^{-4} +10 z^3 a^{-3} -2 a^{-3} z^{-3} -16 z a^{-3} +9 a^{-3} z^{-1} }[/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   \ 0 1 2 3 4 5 6 7 8 9 χ 22                   1 -1 20                 1   1 18               5 1   -4 16             5 1     4 14           7 5       -2 12         8 5         3 10       7 11           4 8     6 4             2 6     7               7 4 4 6                 -2 2 4                   4

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/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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