L9a28

L9a27

L9a29

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a28 at Knotilus!
	L9a28 is [math]\displaystyle{ 9^2_{20} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a28's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,16,7,15 X16,12,17,11 X12,18,13,17 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -9, 4, -8}, {8, -1, 2, -3, 6, -7, 9, -4, 5, -6, 7, -5}

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)^2 t(2)^4-2 t(1)^2 t(2)^3+2 t(1) t(2)^3+2 t(1)^2 t(2)^2-3 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{5}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{5}{q^{5/2}}+q^{3/2}-\frac{5}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{4}{q^{11/2}}-2 \sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^5+4 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-6 a^3 z^5-13 a^3 z^3-12 a^3 z-3 a^3 z^{-1} +a z^5+4 a z^3+4 a z+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^3-a^9 z+2 a^8 z^4-a^8 z^2+3 a^7 z^5-3 a^7 z^3+2 a^7 z+3 a^6 z^6-3 a^6 z^4+a^6 z^2+3 a^5 z^7-7 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^8+2 a^4 z^6-11 a^4 z^4+10 a^4 z^2-3 a^4+5 a^3 z^7-18 a^3 z^5+22 a^3 z^3-14 a^3 z+3 a^3 z^{-1} +a^2 z^8-10 a^2 z^4+12 a^2 z^2-3 a^2+2 a z^7-8 a z^5+9 a z^3-4 a z+a z^{-1} +z^6-4 z^4+4 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                   1 -1 2                 1   1 0               2 1   -1 -2             3 1     2 -4           3 3       0 -6         3 2         1 -8       2 3           1 -10     2 3             -1 -12     2               2 -14 1 2                 -1 -16 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/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.

L9a27 L9a29