L9a47

L9a46

L9a48

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a47 at Knotilus!
	L9a47 is [math]\displaystyle{ 9^3_{2} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a47's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X10,13,5,14 X18,15,11,16 X14,7,15,8 X8,18,9,17 X16,10,17,9 X2536 X4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 5, -6, 7, -3}, {9, -2, 3, -5, 4, -7, 6, -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{-t(3)^2 t(2)^2+t(1) t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-2 t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+t(2)+t(1) t(3)^2-t(3)^2+t(1)-2 t(1) t(3)+t(3)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-8} +3 q^{-7} -5 q^{-6} +8 q^{-5} -8 q^{-4} +10 q^{-3} -7 q^{-2} +q+6 q^{-1} -3 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^8+3 z^2 a^6+a^6 z^{-2} +3 a^6-2 z^4 a^4-4 z^2 a^4-2 a^4 z^{-2} -5 a^4-z^4 a^2+z^2 a^2+a^2 z^{-2} +3 a^2+z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+5 z^2 a^8-2 a^8+3 z^7 a^7-3 z^5 a^7-4 z^3 a^7+3 z a^7+z^8 a^6+8 z^6 a^6-24 z^4 a^6+21 z^2 a^6+a^6 z^{-2} -8 a^6+7 z^7 a^5-10 z^5 a^5+5 z a^5-2 a^5 z^{-1} +z^8 a^4+10 z^6 a^4-26 z^4 a^4+24 z^2 a^4+2 a^4 z^{-2} -9 a^4+4 z^7 a^3-3 z^5 a^3-z^3 a^3+3 z a^3-2 a^3 z^{-1} +5 z^6 a^2-8 z^4 a^2+7 z^2 a^2+a^2 z^{-2} -4 a^2+3 z^5 a-3 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 2   -2 -1               4 1   3 -3             4 3     -1 -5           6 3       3 -7         4 6         2 -9       4 4           0 -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}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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