L11a546

L11a545

L11a547

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Link Presentations

[edit Notes on L11a546's Link Presentations]

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

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 w^2 x-u v w^2+u v w x^2-3 u v w x+3 u v w-u v x^2+3 u v x-3 u v+u w^2 x^2-2 u w^2 x+u w^2-2 u w x^2+4 u w x-2 u w+u x^2-2 u x+u+v w^2 x^2-2 v w^2 x+v w^2-2 v w x^2+4 v w x-2 v w+v x^2-2 v x+v-3 w^2 x^2+3 w^2 x-w^2+3 w x^2-3 w x+w-x^2+x}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{18}{q^{9/2}}+\frac{19}{q^{7/2}}+q^{5/2}-\frac{22}{q^{5/2}}-4 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{11}{q^{11/2}}+9 \sqrt{q}-\frac{15}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-3} +a^9 z^{-1} -3 a^7 z^{-3} -4 a^7 z-6 a^7 z^{-1} +6 a^5 z^3+3 a^5 z^{-3} +12 a^5 z+9 a^5 z^{-1} -3 a^3 z^5-8 a^3 z^3-a^3 z^{-3} -10 a^3 z-4 a^3 z^{-1} -a z^5+a z^3+z^3 a^{-1} +2 a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-5 a^9 z^5+10 a^9 z^3-a^9 z^{-3} -10 a^9 z+5 a^9 z^{-1} +2 a^8 z^8-5 a^8 z^6+10 a^8 z^2+3 a^8 z^{-2} -10 a^8+3 a^7 z^9-5 a^7 z^7-7 a^7 z^5+22 a^7 z^3-3 a^7 z^{-3} -23 a^7 z+12 a^7 z^{-1} +a^6 z^{10}+11 a^6 z^8-40 a^6 z^6+31 a^6 z^4+10 a^6 z^2+6 a^6 z^{-2} -19 a^6+9 a^5 z^9-8 a^5 z^7-34 a^5 z^5+49 a^5 z^3-3 a^5 z^{-3} -27 a^5 z+12 a^5 z^{-1} +a^4 z^{10}+22 a^4 z^8-61 a^4 z^6+41 a^4 z^4+3 a^4 z^{-2} -10 a^4+6 a^3 z^9+12 a^3 z^7-57 a^3 z^5+53 a^3 z^3-a^3 z^{-3} -20 a^3 z+5 a^3 z^{-1} +13 a^2 z^8-17 a^2 z^6+2 a^2 z^4+z^4 a^{-2} +2 a^2 z^2+14 a z^7-21 a z^5+4 z^5 a^{-1} +15 a z^3-z^3 a^{-1} -6 a z+9 z^6-7 z^4+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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   6 1   -5 0                 9 3     6 -2               10 7       -3 -4             12 8         4 -6           11 14           3 -8         7 8             -1 -10       6 13               7 -12     2 5                 -3 -14     6                   6 -16 1 2                     -1 -18 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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