L9a37

L9a36

L9a38

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a37 at Knotilus!
	L9a37 is [math]\displaystyle{ 9^2_{7} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a37's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X18,12,9,11 X2,9,3,10 X4,18,5,17 X16,8,17,7 X14,6,15,5 X6,16,7,15 X8,14,1,13
Gauss code	{1, -4, 2, -5, 7, -8, 6, -9}, {4, -1, 3, -2, 9, -7, 8, -6, 5, -3}

A Braid Representative	{{{braid_table}}}
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^3 v^2-u^3 v+u^2 v^3-4 u^2 v^2+3 u^2 v-u^2-u v^3+3 u v^2-4 u v+u-v^2+v}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-3 q^{15/2}+5 q^{13/2}-7 q^{11/2}+7 q^{9/2}-8 q^{7/2}+6 q^{5/2}-4 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +z a^{-7} -z^5 a^{-5} -2 z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} -z^5 a^{-3} -2 z^3 a^{-3} + a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-10} -z^2 a^{-10} +3 z^5 a^{-9} -4 z^3 a^{-9} +z a^{-9} +4 z^6 a^{-8} -5 z^4 a^{-8} +z^2 a^{-8} +3 z^7 a^{-7} -2 z^5 a^{-7} +z^8 a^{-6} +4 z^6 a^{-6} -7 z^4 a^{-6} +3 z^2 a^{-6} +5 z^7 a^{-5} -9 z^5 a^{-5} +8 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +z^8 a^{-4} +2 z^6 a^{-4} -6 z^4 a^{-4} +4 z^2 a^{-4} - a^{-4} +2 z^7 a^{-3} -3 z^5 a^{-3} +z^3 a^{-3} -2 z a^{-3} + a^{-3} z^{-1} +2 z^6 a^{-2} -5 z^4 a^{-2} +3 z^2 a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +2 z a^{-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   \ -2 -1 0 1 2 3 4 5 6 7 χ 18                   1 -1 16                 2   2 14               3 1   -2 12             4 2     2 10           4 4       0 8         4 3         1 6       2 4           2 4     2 4             -2 2   1 3               2 0   1                 -1 -2 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=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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