L11a511

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 -2 v 2 u 2 - v 2 w 2 u 2 + 2 v w 2 u 2 - w 2 u 2 + 2 v u 2 + 3 v 2 w u 2 -5 v w u 2 + 3 w u 2 - u 2 + v 2 u + v 2 w 2 u -2 v w 2 u + 2 w 2 u -2 v u -3 v 2 w u + 5 v w u -3 w u + u + v w 2 - w 2 + v 2 w -2 v w + w$ (db)
Jones polynomial	$- q 8 + 3 q 7 -6 q 6 + 11 q 5 -14 q 4 + 17 q 3 -16 q 2 + 15 q -10 + 7 q -1 -3 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + z 4 a -2 + 3 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 -3 z 2 a -2 + 5 z 2 a -4 -2 z 2 a -6 -3 z 2 + a 2 -5 a -2 + 4 a -4 - a -6 + 1-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 3 z 9 a -1 + 7 z 9 a -3 + 4 z 9 a -5 + 8 z 8 a -2 + 10 z 8 a -4 + 6 z 8 a -6 + 4 z 8 + 3 a z 7 - z 7 a -1 -13 z 7 a -3 -4 z 7 a -5 + 5 z 7 a -7 + a 2 z 6 -27 z 6 a -2 -35 z 6 a -4 -14 z 6 a -6 + 3 z 6 a -8 -8 z 6 -8 a z 5 -10 z 5 a -1 + 7 z 5 a -3 -2 z 5 a -5 -10 z 5 a -7 + z 5 a -9 -3 a 2 z 4 + 35 z 4 a -2 + 53 z 4 a -4 + 17 z 4 a -6 -6 z 4 a -8 + 2 z 4 + 5 a z 3 + 9 z 3 a -1 + 5 z 3 a -3 + 9 z 3 a -5 + 6 z 3 a -7 -2 z 3 a -9 + 3 a 2 z 2 -29 z 2 a -2 -35 z 2 a -4 -9 z 2 a -6 + 2 z 2 a -8 -2 z 2 -6 z a -1 -8 z a -3 -3 z a -5 - z a -7 - a 2 + 12 a -2 + 10 a -4 + 2 a -6 + 4 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11a511. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$