L9a52

L9a51

L9a53

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a52 at Knotilus!
	L9a52 is [math]\displaystyle{ 9^3_{8} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a52's Link Presentations]

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