L11a381

L11a380

L11a382

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Link Presentations

[edit Notes on L11a381's Link Presentations]

Planar diagram presentation	X12,1,13,2 X20,13,21,14 X10,21,1,22 X14,4,15,3 X18,6,19,5 X6,11,7,12 X22,7,11,8 X4,16,5,15 X8,18,9,17 X16,10,17,9 X2,20,3,19
Gauss code	{1, -11, 4, -8, 5, -6, 7, -9, 10, -3}, {6, -1, 2, -4, 8, -10, 9, -5, 11, -2, 3, -7}

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(2)^2 t(1)^4-t(2) t(1)^4-t(2)^4 t(1)^3+5 t(2)^3 t(1)^3-8 t(2)^2 t(1)^3+5 t(2) t(1)^3-t(1)^3+2 t(2)^4 t(1)^2-9 t(2)^3 t(1)^2+15 t(2)^2 t(1)^2-9 t(2) t(1)^2+2 t(1)^2-t(2)^4 t(1)+5 t(2)^3 t(1)-8 t(2)^2 t(1)+5 t(2) t(1)-t(1)-t(2)^3+t(2)^2}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -10 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{4}{q^{7/2}}-23 q^{5/2}+\frac{10}{q^{5/2}}+26 q^{3/2}-\frac{17}{q^{3/2}}-q^{13/2}+4 q^{11/2}-27 \sqrt{q}+\frac{22}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +2 a z^5-2 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+3 a z^3+3 z^3 a^{-3} -z^3 a^{-5} -a^3 z+3 z a^{-1} -z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +z^2 a^{-6} +9 z^7 a^{-5} -13 z^5 a^{-5} +9 z^3 a^{-5} -3 z a^{-5} +12 z^8 a^{-4} +a^4 z^6-18 z^6 a^{-4} -2 a^4 z^4+12 z^4 a^{-4} +a^4 z^2-4 z^2 a^{-4} +9 z^9 a^{-3} +4 a^3 z^7-3 z^7 a^{-3} -8 a^3 z^5-15 z^5 a^{-3} +5 a^3 z^3+13 z^3 a^{-3} -a^3 z-z a^{-3} - a^{-3} z^{-1} +3 z^{10} a^{-2} +8 a^2 z^8+18 z^8 a^{-2} -17 a^2 z^6-46 z^6 a^{-2} +12 a^2 z^4+33 z^4 a^{-2} -4 a^2 z^2-9 z^2 a^{-2} + a^{-2} +8 a z^9+17 z^9 a^{-1} -11 a z^7-27 z^7 a^{-1} +7 z^5 a^{-1} -2 z^3 a^{-1} +2 a z+5 z a^{-1} - a^{-1} z^{-1} +3 z^{10}+14 z^8-42 z^6+31 z^4-9 z^2 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   7 1   6 8                 10 3     -7 6               13 7       6 4             14 11         -3 2           13 12           1 0         10 15             5 -2       7 12               -5 -4     3 10                 7 -6   1 7                   -6 -8   3                     3 -10 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{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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