L9a43

L9a42

L9a44

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a43 at Knotilus!
	L9a43 is [math]\displaystyle{ 9^3_{4} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a43's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X16,11,17,12 X18,15,9,16 X12,17,13,18 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -4}, {9, -2, 5, -7, 4, -3, 6, -5, 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{t(1) t(3)^3+t(2) t(3)^3-t(3)^3-3 t(1) t(3)^2+2 t(1) t(2) t(3)^2-3 t(2) t(3)^2+2 t(3)^2+3 t(1) t(3)-2 t(1) t(2) t(3)+3 t(2) t(3)-2 t(3)-t(1)+t(1) t(2)-t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-11} +2 q^{-10} -5 q^{-9} +8 q^{-8} -8 q^{-7} +10 q^{-6} -7 q^{-5} +7 q^{-4} -3 q^{-3} + q^{-2} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{12} z^{-2} +4 a^{10} z^{-2} +4 a^{10}-6 a^8 z^2-5 a^8 z^{-2} -11 a^8+3 a^6 z^4+8 a^6 z^2+2 a^6 z^{-2} +7 a^6+a^4 z^4+a^4 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{13} z^5-3 a^{13} z^3+3 a^{13} z-a^{13} z^{-1} +2 a^{12} z^6-4 a^{12} z^4+3 a^{12} z^2+a^{12} z^{-2} -2 a^{12}+2 a^{11} z^7+a^{11} z^5-11 a^{11} z^3+13 a^{11} z-5 a^{11} z^{-1} +a^{10} z^8+6 a^{10} z^6-16 a^{10} z^4+14 a^{10} z^2+4 a^{10} z^{-2} -10 a^{10}+6 a^9 z^7-3 a^9 z^5-16 a^9 z^3+21 a^9 z-9 a^9 z^{-1} +a^8 z^8+10 a^8 z^6-24 a^8 z^4+23 a^8 z^2+5 a^8 z^{-2} -14 a^8+4 a^7 z^7-10 a^7 z^3+11 a^7 z-5 a^7 z^{-1} +6 a^6 z^6-11 a^6 z^4+11 a^6 z^2+2 a^6 z^{-2} -7 a^6+3 a^5 z^5-2 a^5 z^3+a^4 z^4-a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   1 1 -5                 3 1 -2 -7               4     4 -9             3 3     0 -11           7 4       3 -13         4 6         2 -15       4 4           0 -17     1 4             3 -19   1 4               -3 -21   1                 1 -23 1                   -1

Integral Khovanov Homology (db, data source)

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