L9a6

L9a5

L9a7

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a6 at Knotilus!
	L9a6 is [math]\displaystyle{ 9^2_{29} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a6's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X14,7,15,8 X8,15,9,16 X18,13,5,14 X12,17,13,18 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -5, 9, -2, 3, -7, 6, -4, 5, -3, 7, -6}

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{-2 u v^4+3 u v^3-3 u v^2+2 u v-u-v^5+2 v^4-3 v^3+3 v^2-2 v}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{5}{q^{19/2}}-\frac{7}{q^{17/2}}+\frac{7}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{6}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} (-z)-2 a^{11} z^{-1} +3 a^9 z^3+8 a^9 z+5 a^9 z^{-1} -2 a^7 z^5-7 a^7 z^3-7 a^7 z-3 a^7 z^{-1} -a^5 z^5-3 a^5 z^3-2 a^5 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{14}+2 z^2 a^{14}-a^{14}-2 z^5 a^{13}+2 z^3 a^{13}-3 z^6 a^{12}+3 z^4 a^{12}-z^2 a^{12}-3 z^7 a^{11}+4 z^5 a^{11}-6 z^3 a^{11}+6 z a^{11}-2 a^{11} z^{-1} -z^8 a^{10}-5 z^6 a^{10}+14 z^4 a^{10}-15 z^2 a^{10}+5 a^{10}-6 z^7 a^9+15 z^5 a^9-21 z^3 a^9+17 z a^9-5 a^9 z^{-1} -z^8 a^8-4 z^6 a^8+14 z^4 a^8-13 z^2 a^8+5 a^8-3 z^7 a^7+8 z^5 a^7-10 z^3 a^7+9 z a^7-3 a^7 z^{-1} -2 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-2 z a^5 }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 2 1 -1 -8               3     3 -10             3 2     -1 -12           5 3       2 -14         3 4         1 -16       4 4           0 -18     1 3             2 -20   1 4               -3 -22   1                 1 -24 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/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}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/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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