L10a154

L10a153

L10a155

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Link Presentations

[edit Notes on L10a154's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X20,15,17,16 X18,11,19,12 X12,17,13,18 X16,19,5,20 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {6, -5, 7, -4}, {9, -1, 3, -8, 10, -2, 5, -6, 8, -3, 4, -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{(w-1) \left(2 u v w-3 u v+u w^2-3 u w+u+v w^2-3 v w+v-3 w^2+2 w\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q-3+7 q^{-1} -11 q^{-2} +13 q^{-3} -13 q^{-4} +13 q^{-5} -9 q^{-6} +7 q^{-7} -2 q^{-8} + q^{-9} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} z^{-2} -2 a^8 z^{-2} -4 a^8+6 a^6 z^2+a^6 z^{-2} +6 a^6-3 a^4 z^4-5 a^4 z^2-4 a^4-a^2 z^4+2 a^2 z^2+2 a^2+z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+6 z^2 a^{10}+a^{10} z^{-2} -4 a^{10}+2 z^7 a^9-4 z^5 a^9+4 z a^9-2 a^9 z^{-1} +3 z^8 a^8-5 z^6 a^8-z^4 a^8+5 z^2 a^8+2 a^8 z^{-2} -5 a^8+z^9 a^7+9 z^7 a^7-29 z^5 a^7+22 z^3 a^7-4 z a^7-2 a^7 z^{-1} +8 z^8 a^6-11 z^6 a^6-5 z^4 a^6+6 z^2 a^6+a^6 z^{-2} -2 a^6+z^9 a^5+15 z^7 a^5-41 z^5 a^5+36 z^3 a^5-12 z a^5+5 z^8 a^4+z^6 a^4-16 z^4 a^4+14 z^2 a^4-2 a^4+8 z^7 a^3-13 z^5 a^3+12 z^3 a^3-4 z a^3+6 z^6 a^2-7 z^4 a^2+6 z^2 a^2-2 a^2+3 z^5 a-2 z^3 a+z^4-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 χ 3                     1 1 1                   2   -2 -1                 5 1   4 -3               7 3     -4 -5             6 4       2 -7           7 7         0 -9         6 6           0 -11       5 9             4 -13     2 4               -2 -15     5                 5 -17 1 2                   -1 -19 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=-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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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