L6a5

L6a4

L6n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a5 at Knotilus!
	L6a5 is $6_{1}^{3}$ in the Rolfsen table of links. It is a closed three-link chain.

Stained glass window of Trinity symbol, Brazil

French coat of arms.

Russian coat of arms.

Russian passport page-number decoration.

Link Presentations

[edit Notes on L6a5's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X12,7,9,8 X8,11,5,12 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 $u$, $v$, $w$, ...)	${\frac {t(2)t(1)+t(3)t(1)-t(1)-t(2)+t(2)t(3)-t(3)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$q^{-1}-2q^{-2}+3q^{-3}-q^{-4}+3q^{-5}-q^{-6}+q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}-2a^{6}z^{-2}-3a^{6}+2a^{4}z^{2}+a^{4}z^{-2}+3a^{4}+a^{2}z^{2}$ (db)
Kauffman polynomial	$z^{4}a^{8}-3z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{5}a^{7}-z^{3}a^{7}-3za^{7}+2a^{7}z^{-1}+4z^{4}a^{6}-9z^{2}a^{6}-2a^{6}z^{-2}+5a^{6}+z^{5}a^{5}+z^{3}a^{5}-3za^{5}+2a^{5}z^{-1}+3z^{4}a^{4}-5z^{2}a^{4}-a^{4}z^{-2}+3a^{4}+2z^{3}a^{3}+z^{2}a^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 χ -1             1 1 -3           2 1 -1 -5         1     1 -7         2     2 -9     3 1       2 -11   1 3         2 -13               0 -15 1             1

Integral Khovanov Homology (db, data source)

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

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.

L6a4 L6n1