L6a1

L5a1

L6a2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a1 at Knotilus!
	L6a1 is [math]\displaystyle{ 6^2_3 }[/math] in the Rolfsen table of links.

A kolam with two cycles/components[1]	Depiction with two eights interlaced	Mongolian ornament ; the two eights are horizontal
Another one, sum of two L6a1	Another depiction

Link Presentations

[edit Notes on L6a1's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X12,8,5,7 X8,12,9,11 X2536 X4,9,1,10
Gauss code	{1, -5, 2, -6}, {5, -1, 3, -4, 6, -2, 4, -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 v-2 u-2 v+1}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{5/2}}-q^{3/2}+\frac{2}{q^{3/2}}+2 \sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^{-1} -2 z a^3-a^3 z^{-1} +z^3 a+z a-z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^3-2 a^5 z+a^5 z^{-1} +a^4 z^4-a^4+a^3 z^5-a^3 z+a^3 z^{-1} +3 a^2 z^4-3 a^2 z^2+a z^5+z^3 a^{-1} -z a^{-1} +2 z^4-3 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   \ -4 -3 -2 -1 0 1 2 χ 4             1 1 2           1   -1 0         1 1   0 -2       2 2     0 -4     1         1 -6     2         2 -8 1 1           0 -10 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=0 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{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=2 }[/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.

L5a1 L6a2