L11a408

L11a407

L11a409

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Link Presentations

[edit Notes on L11a408's Link Presentations]

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

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(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+2 t(1) t(2)^2-3 t(1) t(3) t(2)^2+5 t(3) t(2)^2-2 t(2)^2-3 t(1) t(3)^2 t(2)+5 t(3)^2 t(2)-5 t(1) t(2)+8 t(1) t(3) t(2)-8 t(3) t(2)+3 t(2)+2 t(1) t(3)^2-2 t(3)^2+2 t(1)-5 t(1) t(3)+3 t(3)-1}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-17 q+21-19 q^{-1} +17 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6-3 z^2 a^4-2 a^4+3 z^4 a^2+4 z^2 a^2+a^2 z^{-2} +3 a^2-z^6-2 z^4-5 z^2-2 z^{-2} -4+2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} -z^2 a^{-4} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-8 a^5 z^5+z^5 a^{-5} +7 a^5 z^3-z^3 a^{-5} -2 a^5 z+4 a^4 z^8-6 a^4 z^6+4 z^6 a^{-4} -2 a^4 z^4-6 z^4 a^{-4} +5 a^4 z^2+3 z^2 a^{-4} -a^4+3 a^3 z^9+3 a^3 z^7+7 z^7 a^{-3} -19 a^3 z^5-10 z^5 a^{-3} +18 a^3 z^3+4 z^3 a^{-3} -6 a^3 z+a^2 z^{10}+10 a^2 z^8+7 z^8 a^{-2} -20 a^2 z^6-5 z^6 a^{-2} +5 a^2 z^4-5 z^4 a^{-2} +6 a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -3 a^2-4 a^{-2} +7 a z^9+4 z^9 a^{-1} +7 z^7 a^{-1} -20 a z^5-20 z^5 a^{-1} +14 a z^3+8 z^3 a^{-1} +4 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +z^{10}+13 z^8-22 z^6+5 z^4+7 z^2+2 z^{-2} -6 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               10 7       -3 1             11 7         4 -1           9 11           2 -3         8 10             -2 -5       4 9               5 -7     3 8                 -5 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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