L6a5

L6a4

L6n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a5 at Knotilus!
	L6a5 is $6^3_1$ 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
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	$q^{-1} -2 q^{-2} +3 q^{-3} - q^{-4} +3 q^{-5} - q^{-6} + q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^8 z^{-2} -2 a^6 z^{-2} -3 a^6+2 a^4 z^2+a^4 z^{-2} +3 a^4+a^2 z^2$ (db)
Kauffman polynomial	$z^4 a^8-3 z^2 a^8-a^8 z^{-2} +3 a^8+z^5 a^7-z^3 a^7-3 z a^7+2 a^7 z^{-1} +4 z^4 a^6-9 z^2 a^6-2 a^6 z^{-2} +5 a^6+z^5 a^5+z^3 a^5-3 z a^5+2 a^5 z^{-1} +3 z^4 a^4-5 z^2 a^4-a^4 z^{-2} +3 a^4+2 z^3 a^3+z^2 a^2$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ 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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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