L9a31

L9a30

L9a32

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a31 at Knotilus!
	L9a31 is [math]\displaystyle{ 9^2_{39} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a31's Link Presentations]

Planar diagram presentation	X8192 X16,9,17,10 X6718 X18,13,7,14 X10,4,11,3 X14,6,15,5 X4,12,5,11 X12,17,13,18 X2,16,3,15
Gauss code	{1, -9, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}

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{\left(v^2-v+1\right) (u v-u+1) (u v-v+1)}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+3 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-3 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^7+5 a z^5-z^5 a^{-1} +9 a z^3-3 z^3 a^{-1} +7 a z-3 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4-a^6 z^2+3 a^5 z^5-3 a^5 z^3+5 a^4 z^6-7 a^4 z^4+4 a^4 z^2-a^4+5 a^3 z^7-8 a^3 z^5+z^5 a^{-3} +8 a^3 z^3-2 z^3 a^{-3} -4 a^3 z+z a^{-3} +a^3 z^{-1} +2 a^2 z^8+4 a^2 z^6+3 z^6 a^{-2} -12 a^2 z^4-6 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} -3 a^2+9 a z^7+4 z^7 a^{-1} -20 a z^5-8 z^5 a^{-1} +18 a z^3+5 z^3 a^{-1} -9 a z-4 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +2 z^8+2 z^6-10 z^4+7 z^2-3 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   1 1 6                 2   -2 4               4 1   3 2             4 3     -1 0           6 3       3 -2         4 5         1 -4       4 5           -1 -6     2 4             2 -8   1 4               -3 -10   2                 2 -12 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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