L9a51

L9a50

L9a52

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a51 at Knotilus!
	L9a51 is [math]\displaystyle{ 9^3_{11} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a51's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X8,12,9,11 X18,8,11,7 X16,13,17,14 X14,6,15,5 X10,16,5,15 X2,9,3,10 X4,18,1,17
Gauss code	{1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -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{u v^2 w^2-2 u v^2 w-2 u v w^2+4 u v w-2 u v+u w^2-2 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+2 v+2 w-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+7 q^5-8 q^4+11 q^3-10 q^2- q^{-2} +9 q+4 q^{-1} -6 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-6} + a^{-6} z^{-2} + a^{-6} -2 z^4 a^{-4} -4 z^2 a^{-4} -2 a^{-4} z^{-2} -3 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} z^{-2} + a^{-2} -z^4-z^2+1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -10 z^4 a^{-6} +10 z^2 a^{-6} + a^{-6} z^{-2} -5 a^{-6} +5 z^7 a^{-5} -3 z^5 a^{-5} -4 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +2 z^8 a^{-4} +8 z^6 a^{-4} -21 z^4 a^{-4} +17 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +10 z^7 a^{-3} -16 z^5 a^{-3} +2 z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +2 z^8 a^{-2} +6 z^6 a^{-2} -18 z^4 a^{-2} +9 z^2 a^{-2} + a^{-2} z^{-2} -3 a^{-2} +5 z^7 a^{-1} +a z^5-9 z^5 a^{-1} -a z^3+3 z^3 a^{-1} +4 z^6-8 z^4+3 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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 15                   1 1 13                 3 1 -2 11               4     4 9             4 3     -1 7           7 4       3 5         4 5         1 3       5 6           -1 1     3 6             3 -1   1 3               -2 -3   3                 3 -5 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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