L6a2

L6a1

L6a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a2 at Knotilus!
	L6a2 is $6_{2}^{2}$ in the Rolfsen table of links.

Mongolian ornament (4 crossings are unnecessary)

Link Presentations

[edit Notes on L6a2's Link Presentations]

Planar diagram presentation	X8192 X12,5,7,6 X10,3,11,4 X4,11,5,12 X2738 X6,9,1,10
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 4, -2}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}(-v)-uv^{2}+uv-u-v}{uv}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z+a^{7}z^{-1}-a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$-z^{3}a^{9}+2za^{9}-z^{4}a^{8}+z^{2}a^{8}-z^{5}a^{7}+2z^{3}a^{7}-3za^{7}+a^{7}z^{-1}-2z^{4}a^{6}+2z^{2}a^{6}-a^{6}-z^{5}a^{5}+2z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+2za^{3}$ (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   \ -6 -5 -4 -3 -2 -1 0 χ -2             1 1 -4           1 1 0 -6         1     1 -8       1 1     0 -10     1 1       0 -12     1         1 -14 1 1           0 -16 1             1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L6a1 L6a3