L9a42

L9a41

L9a43

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a42 at Knotilus!
	L9a42 is [math]\displaystyle{ 9^2_{41} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a42's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X18,5,9,6 X6,9,7,10 X16,12,17,11 X14,8,15,7 X4,14,5,13 X8,16,1,15 X2,17,3,18
Gauss code	{1, -9, 2, -7, 3, -4, 6, -8}, {4, -1, 5, -2, 7, -6, 8, -5, 9, -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-1) (v-1) \left(u^2 v^2-u v^2+3 u v-u+1\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+6 q^{7/2}-\frac{1}{q^{7/2}}-9 q^{5/2}+\frac{3}{q^{5/2}}+9 q^{3/2}-\frac{6}{q^{3/2}}+q^{11/2}-10 \sqrt{q}+\frac{8}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +a z^5-5 z^5 a^{-1} +z^5 a^{-3} +3 a z^3-9 z^3 a^{-1} +3 z^3 a^{-3} +3 a z-6 z a^{-1} +3 z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +5 z^6 a^{-4} -6 z^4 a^{-4} +3 z^2 a^{-4} +5 z^7 a^{-3} +a^3 z^5-6 z^5 a^{-3} -2 a^3 z^3+4 z^3 a^{-3} +a^3 z-2 z a^{-3} +2 z^8 a^{-2} +3 a^2 z^6+5 z^6 a^{-2} -6 a^2 z^4-12 z^4 a^{-2} +3 a^2 z^2+6 z^2 a^{-2} +4 a z^7+9 z^7 a^{-1} -7 a z^5-17 z^5 a^{-1} +3 a z^3+12 z^3 a^{-1} -2 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^8+3 z^6-11 z^4+5 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   1 -1 10                 2   2 8               4 1   -3 6             5 2     3 4           4 4       0 2         6 5         1 0       4 6           2 -2     2 4             -2 -4   1 4               3 -6   2                 -2 -8 1                   1

Integral Khovanov Homology (db, data source)

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