L7a7

L7a6

L7n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a7 at Knotilus!
	L7a7 is $7^3_1$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L7a7's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,12,9,11 X8,14,5,13 X12,8,13,7 X2536 X4,9,1,10
Gauss code	{1, -6, 2, -7}, {6, -1, 5, -4}, {7, -2, 3, -5, 4, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v w+u (-v)-u w+2 u-2 v w+v+w-1}{\sqrt{u} \sqrt{v} \sqrt{w}}$ (db)
Jones polynomial	$q^{-4} -q^3- q^{-3} +3 q^2+4 q^{-2} -3 q-3 q^{-1} +4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^4 z^{-2} +a^4-2 z^2 a^2-2 a^2 z^{-2} -3 a^2+z^4+2 z^2+ z^{-2} +2-z^2 a^{-2}$ (db)
Kauffman polynomial	$a^2 z^6+z^6+a^3 z^5+4 a z^5+3 z^5 a^{-1} +a^4 z^4+a^2 z^4+3 z^4 a^{-2} +3 z^4-4 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -3 a^4 z^2-5 a^2 z^2-3 z^2 a^{-2} -5 z^2-3 a^3 z-3 a z+3 a^4+5 a^2+3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 χ 7               1 -1 5             2   2 3           1 1   0 1         3 2     1 -1       3 4       1 -3     1           1 -5     3           3 -7 1 1             0 -9 1               1

Integral Khovanov Homology (db, data source)

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

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.

L7a6 L7n1