L11a440

L11a439

L11a441

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Link Presentations

[edit Notes on L11a440's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X12,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X22,19,13,20 X20,12,21,11 X10,22,11,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -9, 8, -3}, {11, -2, 3, -5, 4, -6, 7, -8, 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^2-3 u v^2 w+u v^2+u v w^3-3 u v w^2+4 u v w-2 u v+u w^2-2 u w+u+v^3 \left(-w^3\right)+2 v^3 w^2-v^3 w+2 v^2 w^3-4 v^2 w^2+3 v^2 w-v^2-v w^3+3 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^{-11} +3 q^{-10} -6 q^{-9} +10 q^{-8} -13 q^{-7} +17 q^{-6} -15 q^{-5} +14 q^{-4} -10 q^{-3} +7 q^{-2} -3 q^{-1} +1 }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{10}-2 a^{10}+3 z^4 a^8+9 z^2 a^8+a^8 z^{-2} +6 a^8-2 z^6 a^6-8 z^4 a^6-12 z^2 a^6-2 a^6 z^{-2} -9 a^6-z^6 a^4-z^4 a^4+5 z^2 a^4+a^4 z^{-2} +5 a^4+z^4 a^2+2 z^2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{13} z^5-2 a^{13} z^3+a^{13} z+3 a^{12} z^6-6 a^{12} z^4+3 a^{12} z^2-a^{12}+4 a^{11} z^7-5 a^{11} z^5-a^{11} z^3+a^{11} z+4 a^{10} z^8-3 a^{10} z^6-3 a^{10} z^4+2 a^{10} z^2+3 a^9 z^9-7 a^9 z^5+9 a^9 z^3-3 a^9 z+a^8 z^{10}+8 a^8 z^8-27 a^8 z^6+38 a^8 z^4-23 a^8 z^2-a^8 z^{-2} +9 a^8+7 a^7 z^9-15 a^7 z^7+9 a^7 z^5+7 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+9 a^6 z^8-37 a^6 z^6+54 a^6 z^4-38 a^6 z^2-2 a^6 z^{-2} +13 a^6+4 a^5 z^9-8 a^5 z^7+2 a^5 z^5+3 a^5 z^3-5 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-15 a^4 z^6+16 a^4 z^4-14 a^4 z^2-a^4 z^{-2} +6 a^4+3 a^3 z^7-8 a^3 z^5+4 a^3 z^3+a^2 z^6-3 a^2 z^4+2 a^2 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     2   -2 -3                   5 1   4 -5                 6 3     -3 -7               8 4       4 -9             7 6         -1 -11           10 8           2 -13         6 10             4 -15       4 7               -3 -17     2 6                 4 -19   1 4                   -3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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