L11a437

L11a436

L11a438

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Link Presentations

[edit Notes on L11a437's Link Presentations]

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

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{(v-1) (w-1) \left(3 u v-2 u+2 v^2-3 v\right)}{\sqrt{u} v^{3/2} \sqrt{w}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-10} +3 q^{-9} -6 q^{-8} +9 q^{-7} -10 q^{-6} +13 q^{-5} -12 q^{-4} +11 q^{-3} -7 q^{-2} +q+5 q^{-1} -2 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+3 a^8 z^2+2 a^8-2 a^6 z^4-2 a^6 z^2+a^6 z^{-2} -2 a^4 z^4-2 a^4 z^2-2 a^4 z^{-2} -3 a^4-a^2 z^4+a^2 z^{-2} +a^2+z^2+1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{11}-4 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-12 z^6 a^{10}+14 z^4 a^{10}-7 z^2 a^{10}+2 a^{10}+3 z^9 a^9-8 z^7 a^9-2 z^5 a^9+11 z^3 a^9-4 z a^9+z^{10} a^8+5 z^8 a^8-30 z^6 a^8+38 z^4 a^8-18 z^2 a^8+4 a^8+6 z^9 a^7-17 z^7 a^7+10 z^5 a^7-z^3 a^7+z^{10} a^6+5 z^8 a^6-21 z^6 a^6+19 z^4 a^6-3 z^2 a^6+a^6 z^{-2} -3 a^6+3 z^9 a^5-5 z^7 a^5+6 z^5 a^5-10 z^3 a^5+8 z a^5-2 a^5 z^{-1} +3 z^8 a^4-9 z^4 a^4+13 z^2 a^4+2 a^4 z^{-2} -8 a^4+3 z^7 a^3-5 z^3 a^3+6 z a^3-2 a^3 z^{-1} +3 z^6 a^2-3 z^4 a^2+3 z^2 a^2+a^2 z^{-2} -3 a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     1   -1 -1                   4 1   3 -3                 5 3     -2 -5               6 2       4 -7             6 5         -1 -9           7 6           1 -11         5 8             3 -13       4 5               -1 -15     2 5                 3 -17   1 4                   -3 -19   2                     2 -21 1                       -1

Integral Khovanov Homology (db, data source)

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