L11n345

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_11,19,12,18 X_16,8,17,7 X_17,21,18,20 X_19,5,20,12 X_8,22,9,21 X_22,10,13,9 X_10,14,11,13 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, -3, 6}, {9, -2, 11, -4, -5, 3, -6, 5, 7, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 3 - v 3 w u 3 + v 2 w u 3 - v 3 u 2 + v u 2 - v 2 w u + w u - v - w + 1$ (db)
Jones polynomial	$q 10 - q 8 + q 7 - q 6 + 3 q 5 -2 q 4 + 3 q 3 - q 2 + q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$- z 8 a -6 + z 6 a -4 -7 z 6 a -6 + z 6 a -8 + 6 z 4 a -4 -17 z 4 a -6 + 6 z 4 a -8 + 11 z 2 a -4 -20 z 2 a -6 + 9 z 2 a -8 - z 2 a -10 + 7 a -4 -11 a -6 + 4 a -8 + a -4 z -2 -2 a -6 z -2 + a -8 z -2$ (db)
Kauffman polynomial	$z 9 a -5 + z 9 a -7 + z 8 a -4 + 4 z 8 a -6 + 3 z 8 a -8 -5 z 7 a -5 -3 z 7 a -7 + 2 z 7 a -9 -7 z 6 a -4 -25 z 6 a -6 -18 z 6 a -8 + 4 z 5 a -5 -7 z 5 a -7 -11 z 5 a -9 + 17 z 4 a -4 + 48 z 4 a -6 + 29 z 4 a -8 - z 4 a -10 + z 4 a -12 + 8 z 3 a -5 + 20 z 3 a -7 + 11 z 3 a -9 - z 3 a -11 -18 z 2 a -4 -37 z 2 a -6 -16 z 2 a -8 - z 2 a -10 -4 z 2 a -12 -11 z a -5 -11 z a -7 + 8 a -4 + 13 a -6 + 5 a -8 + a -12 + 2 a -5 z -1 + 2 a -7 z -1 - a -4 z -2 -2 a -6 z -2 - a -8 z -2$ (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 =

6 is the signature of L11n345. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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