L11a456

L11a455

L11a457

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Link Presentations

[edit Notes on L11a456's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X20,11,21,12 X18,8,19,7 X16,10,17,9 X8,18,9,17 X22,19,13,20 X10,14,11,13 X12,21,5,22 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 5, -8, 3, -9}, {8, -2, 11, -5, 6, -4, 7, -3, 9, -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{u v^3 w^2-u v^3 w+u v^2 w^3-4 u v^2 w^2+5 u v^2 w-u v^2-2 u v w^3+5 u v w^2-4 u v w+u v+u w^3-2 u w^2+u w-v^3 w^2+2 v^3 w-v^3-v^2 w^3+4 v^2 w^2-5 v^2 w+2 v^2+v w^3-5 v w^2+4 v w-v+w^2-w}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-3 q^5- q^{-5} +7 q^4+3 q^{-4} -11 q^3-7 q^{-3} +17 q^2+13 q^{-2} -18 q-16 q^{-1} +19 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^4 a^{-4} -a^4 z^2+2 z^2 a^{-4} + a^{-4} z^{-2} -a^4+2 a^{-4} -z^6 a^{-2} +2 a^2 z^4-3 z^4 a^{-2} +3 a^2 z^2-6 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2-5 a^{-2} -z^6-z^4+z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-6} -3 z^4 a^{-6} +2 z^2 a^{-6} +3 z^7 a^{-5} +a^5 z^5-8 z^5 a^{-5} -2 a^5 z^3+5 z^3 a^{-5} +a^5 z+5 z^8 a^{-4} +3 a^4 z^6-14 z^6 a^{-4} -5 a^4 z^4+16 z^4 a^{-4} +3 a^4 z^2-13 z^2 a^{-4} - a^{-4} z^{-2} -a^4+6 a^{-4} +4 z^9 a^{-3} +5 a^3 z^7-5 z^7 a^{-3} -6 a^3 z^5-3 z^5 a^{-3} +a^3 z^3+6 z^3 a^{-3} +a^3 z-5 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +6 a^2 z^8+12 z^8 a^{-2} -7 a^2 z^6-41 z^6 a^{-2} +3 a^2 z^4+55 z^4 a^{-2} -a^2 z^2-39 z^2 a^{-2} -2 a^{-2} z^{-2} +13 a^{-2} +4 a z^9+8 z^9 a^{-1} +2 a z^7-11 z^7 a^{-1} -12 a z^5+10 a z^3+8 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +z^{10}+13 z^8-36 z^6+44 z^4-28 z^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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     2   -2 9                   5 1   4 7                 7 3     -4 5               10 4       6 3             8 7         -1 1           11 10           1 -1         8 11             3 -3       5 8               -3 -5     2 8                 6 -7   1 5                   -4 -9   2                     2 -11 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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