L11a518

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_22,9,13,10 X_20,11,21,12 X_12,19,7,20 X_10,21,11,22 X_18,14,19,13 X_16,6,17,5 X₂₇₃₈ X_4,16,5,15 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -6, 4, -5}, {7, -2, 10, -8, 11, -7, 5, -4, 6, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 3 - v u 3 - v 2 w u 3 + v w u 3 - v 2 u 2 - v 2 w 2 u 2 + v w 2 u 2 + 2 v u 2 + 2 v 2 w u 2 -2 v w u 2 + w u 2 - u 2 + v 2 w 2 u -2 v w 2 u + w 2 u - v u - v 2 w u + 2 v w u -2 w u + u + v w 2 -2 w 2 - v w + w$ (db)
Jones polynomial	$- q 6 + 3 q 5 -5 q 4 + 8 q 3 -9 q 2 + 10 q -9 + 8 q -1 -5 q -2 + 4 q -3 - q -4 + q -5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 -2 a 2 z 4 + 3 z 4 a -2 - z 4 a -4 + 3 z 4 + a 4 z 2 -7 a 2 z 2 + 2 z 2 a -2 -2 z 2 a -4 + 2 z 2 + 3 a 4 -6 a 2 + a -2 + 2 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a 2 z 10 + z 10 + a 3 z 9 + 4 a z 9 + 3 z 9 a -1 + a 4 z 8 -3 a 2 z 8 + 5 z 8 a -2 + z 8 -4 a 3 z 7 -16 a z 7 -5 z 7 a -1 + 7 z 7 a -3 -7 a 4 z 6 -3 a 2 z 6 -8 z 6 a -2 + 7 z 6 a -4 -11 z 6 + a 3 z 5 + 15 a z 5 -4 z 5 a -1 -13 z 5 a -3 + 5 z 5 a -5 + 17 a 4 z 4 + 17 a 2 z 4 -5 z 4 a -2 -10 z 4 a -4 + 3 z 4 a -6 + 8 z 4 + 10 a 3 z 3 + 4 a z 3 + 3 z 3 a -1 + 5 z 3 a -3 -3 z 3 a -5 + z 3 a -7 -17 a 4 z 2 -20 a 2 z 2 + 7 z 2 a -2 + 4 z 2 a -4 - z 2 a -6 - z 2 -9 a 3 z -9 a z + 7 a 4 + 11 a 2 -2 a -2 + 3 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11a518. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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