L11n316

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,17,8,16 X_20,10,21,9 X_18,12,19,11 X_22,20,11,19 X_15,9,16,8 X_10,22,5,21 X_17,14,18,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, 4, -8}, {5, -2, 11, 9, -7, 3, -9, -5, 6, -4, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 2 - v w u 2 + w u 2 - u 2 + v 2 u -3 v u -2 v 2 w u + 3 v w u - w u + 2 u - v 2 + v + v 2 w -2 v w$ (db)
Jones polynomial	$q 9 -3 q 8 + 5 q 7 -6 q 6 + 8 q 5 -7 q 4 + 7 q 3 -4 q 2 + 3 q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2 z 4 a -4 - z 4 a -6 + 3 z 2 a -2 -5 z 2 a -4 + z 2 a -8 + 4 a -2 -6 a -4 + 2 a -6 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$z 9 a -5 + z 9 a -7 + z 8 a -4 + 4 z 8 a -6 + 3 z 8 a -8 -2 z 7 a -5 + z 7 a -7 + 3 z 7 a -9 - z 6 a -4 -11 z 6 a -6 -9 z 6 a -8 + z 6 a -10 + 3 z 5 a -3 + 5 z 5 a -5 -8 z 5 a -7 -10 z 5 a -9 - z 4 a -4 + 8 z 4 a -6 + 6 z 4 a -8 -3 z 4 a -10 -4 z 3 a -3 -8 z 3 a -5 + 3 z 3 a -7 + 7 z 3 a -9 + 5 z 2 a -2 + 10 z 2 a -4 -3 z 2 a -8 + 2 z 2 a -10 + 6 z a -3 + 6 z a -5 -5 a -2 -8 a -4 -3 a -6 + a -8 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

2 is the signature of L11n316. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = 0$	${\mathbb Z}^{3}$	${\mathbb Z}^{2}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 8$	${\mathbb Z}_2$	${\mathbb Z}$