L10a171

L10a170

L10a172

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Link Presentations

[edit Notes on L10a171's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X10,13,5,14 X20,16,17,15 X14,7,15,8 X16,20,11,19 X8,17,9,18 X18,9,19,10 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {7, -8, 6, -4}, {9, -1, 5, -7, 8, -3}, {10, -2, 3, -5, 4, -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{u v w^2 x-u v w^2+u v w x^2-3 u v w x+2 u v w+u v x-u v-u w^2 x-u w x^2+3 u w x-u w+u x^2-2 u x+u+v w^2 x^2-2 v w^2 x+v w^2-v w x^2+3 v w x-v w-v x-w^2 x^2+w^2 x+2 w x^2-3 w x+w-x^2+x}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^9+a^9 z^{-1} +a^9 z^{-3} -3 z^3 a^7-6 z a^7-6 a^7 z^{-1} -3 a^7 z^{-3} +2 z^5 a^5+6 z^3 a^5+10 z a^5+9 a^5 z^{-1} +3 a^5 z^{-3} +z^5 a^3-4 z a^3-4 a^3 z^{-1} -a^3 z^{-3} -z^3 a-z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-z^2 a^{10}-5 z^7 a^9+10 z^5 a^9-10 z^3 a^9+11 z a^9-5 a^9 z^{-1} +a^9 z^{-3} -4 z^8 a^8+2 z^6 a^8+9 z^4 a^8-14 z^2 a^8-3 a^8 z^{-2} +10 a^8-z^9 a^7-13 z^7 a^7+42 z^5 a^7-54 z^3 a^7+33 z a^7-12 a^7 z^{-1} +3 a^7 z^{-3} -8 z^8 a^6+10 z^6 a^6+8 z^4 a^6-26 z^2 a^6-6 a^6 z^{-2} +19 a^6-z^9 a^5-13 z^7 a^5+42 z^5 a^5-54 z^3 a^5+33 z a^5-12 a^5 z^{-1} +3 a^5 z^{-3} -4 z^8 a^4+2 z^6 a^4+9 z^4 a^4-14 z^2 a^4-3 a^4 z^{-2} +10 a^4-5 z^7 a^3+10 z^5 a^3-10 z^3 a^3+11 z a^3-5 a^3 z^{-1} +a^3 z^{-3} -3 z^6 a^2+5 z^4 a^2-z^2 a^2-z^5 a+2 z^3 a-z a }[/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 χ 2                     1 1 0                   2   -2 -2                 5 1   4 -4               6 4     -2 -6             8 3       5 -8           6 8         2 -10         8 6           2 -12       3 8             5 -14     4 6               -2 -16   1 5                 4 -18   2                   -2 -20 1                     1

Integral Khovanov Homology (db, data source)

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