L6a1

L5a1

L6a2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a1 at Knotilus!
	L6a1 is $6_{3}^{2}$ 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
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv-2u-2v+1}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{5/2}}}-q^{3/2}+{\frac {2}{q^{3/2}}}+2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-2za^{3}-a^{3}z^{-1}+z^{3}a+za-za^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+a^{4}z^{4}-a^{4}+a^{3}z^{5}-a^{3}z+a^{3}z^{-1}+3a^{2}z^{4}-3a^{2}z^{2}+az^{5}+z^{3}a^{-1}-za^{-1}+2z^{4}-3z^{2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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