L9a32

L9a31

L9a33

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a32 at Knotilus!
	L9a32 is [math]\displaystyle{ 9^2_{40} }[/math] in the Rolfsen table of links.

A traditional symbol of the Christian Trinity (a triquetra interlaced with a circle, or "Trinity knot")		Symmetric version, with three lines touching at center		Symmetric version, with three lines touching at cicrumference
Logo of the Canadian Undergraduate Mathematics Conference		Circled hypotrochoid

Link Presentations

[edit Notes on L9a32's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X18,13,7,14 X14,9,15,10 X10,17,11,18 X16,5,17,6 X2738 X4,11,5,12 X6,15,1,16
Gauss code	{1, -7, 2, -8, 6, -9}, {7, -1, 4, -5, 8, -2, 3, -4, 9, -6, 5, -3}

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(2)^4-3 t(1) t(2)^3+3 t(2)^3-3 t(1)^2 t(2)^2+5 t(1) t(2)^2-3 t(2)^2+3 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{9}{q^{11/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -3 a^7 z^3-4 a^7 z-2 a^7 z^{-1} -3 a^5 z^3-3 a^5 z-a^3 z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-10 a^{11} z^5+9 a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +2 a^{10} z^8-a^{10} z^6-12 a^{10} z^4+14 a^{10} z^2-3 a^{10}+9 a^9 z^7-26 a^9 z^5+22 a^9 z^3-11 a^9 z+3 a^9 z^{-1} +2 a^8 z^8+5 a^8 z^6-21 a^8 z^4+14 a^8 z^2-3 a^8+6 a^7 z^7-10 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +7 a^6 z^6-9 a^6 z^4+3 a^6 z^2+6 a^5 z^5-6 a^5 z^3+3 a^5 z+3 a^4 z^4+a^3 z^3 }[/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 -2                   1 1 -4                 3 1 -2 -6               3     3 -8             4 3     -1 -10           5 3       2 -12         3 4         1 -14       4 5           -1 -16     2 4             2 -18   1 3               -2 -20   2                 2 -22 1                   -1

Integral Khovanov Homology (db, data source)

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