L11a422

L11a421

L11a423

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Link Presentations

[edit Notes on L11a422's Link Presentations]

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

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(3)^4+t(1) t(3)^4-t(1) t(2) t(3)^4+2 t(2) t(3)^4-t(3)^4-t(1) t(2)^2 t(3)^3+2 t(2)^2 t(3)^3-3 t(1) t(3)^3+5 t(1) t(2) t(3)^3-5 t(2) t(3)^3+2 t(3)^3+2 t(1) t(2)^2 t(3)^2-3 t(2)^2 t(3)^2+3 t(1) t(3)^2-6 t(1) t(2) t(3)^2+6 t(2) t(3)^2-2 t(3)^2-2 t(1) t(2)^2 t(3)+3 t(2)^2 t(3)-2 t(1) t(3)+5 t(1) t(2) t(3)-5 t(2) t(3)+t(3)+t(1) t(2)^2-t(2)^2+t(1)-2 t(1) t(2)+t(2)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-4 q^{-1} +10 q^{-2} -14 q^{-3} +21 q^{-4} -22 q^{-5} +23 q^{-6} -19 q^{-7} +14 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{10}-a^{10} z^{-2} -2 a^{10}+3 z^4 a^8+8 z^2 a^8+4 a^8 z^{-2} +9 a^8-2 z^6 a^6-7 z^4 a^6-13 z^2 a^6-5 a^6 z^{-2} -14 a^6-z^6 a^4+6 z^2 a^4+2 a^4 z^{-2} +7 a^4+z^4 a^2+z^2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-9 z^5 a^{11}+7 z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} +8 z^8 a^{10}-13 z^6 a^{10}+14 z^4 a^{10}-11 z^2 a^{10}-a^{10} z^{-2} +5 a^{10}+6 z^9 a^9-20 z^5 a^9+33 z^3 a^9-21 z a^9+5 a^9 z^{-1} +2 z^{10} a^8+16 z^8 a^8-51 z^6 a^8+65 z^4 a^8-44 z^2 a^8-4 a^8 z^{-2} +18 a^8+13 z^9 a^7-20 z^7 a^7-7 z^5 a^7+34 z^3 a^7-29 z a^7+9 a^7 z^{-1} +2 z^{10} a^6+16 z^8 a^6-56 z^6 a^6+67 z^4 a^6-47 z^2 a^6-5 a^6 z^{-2} +21 a^6+7 z^9 a^5-10 z^7 a^5-5 z^5 a^5+13 z^3 a^5-13 z a^5+5 a^5 z^{-1} +8 z^8 a^4-20 z^6 a^4+18 z^4 a^4-14 z^2 a^4-2 a^4 z^{-2} +9 a^4+4 z^7 a^3-8 z^5 a^3+3 z^3 a^3+z^6 a^2-2 z^4 a^2+z^2 a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     3   -3 -3                   7 1   6 -5                 8 4     -4 -7               13 6       7 -9             11 10         -1 -11           12 11           1 -13         8 12             4 -15       6 11               -5 -17     2 8                 6 -19   1 6                   -5 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/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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