L11a451

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_20,10,21,9 X_16,8,17,7 X_18,12,19,11 X_22,16,13,15 X_12,18,5,17 X_10,20,11,19 X_8,22,9,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8, 5, -7}, {11, -2, 6, -4, 7, -5, 8, -3, 9, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 3 - v w u 3 + w u 3 - u 3 + 2 v 2 u 2 -3 v u 2 - v 2 w u 2 + 3 v w u 2 -2 w u 2 + u 2 + 2 v 3 u -3 v 2 u + v u - v 3 w u + 3 v 2 w u -2 v w u - v 3 + v 2 + v 3 w -2 v 2 w$ (db)
Jones polynomial	$- q 7 + 3 q 6 -4 q 5 + 7 q 4 -8 q 3 + 11 q 2 -10 q + 9-7 q -1 + 5 q -2 -2 q -3 + q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 4 -2 z 2 a 2 + z 4 - z 2 + z -2 + z 4 a -2 -2 z 2 a -2 -2 a -2 z -2 -3 a -2 + z 4 a -4 + z 2 a -4 + a -4 z -2 + 2 a -4 - z 2 a -6$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 5 z 9 a -3 + 3 z 9 a -5 + 2 z 8 a -2 + 2 z 8 a -4 + 3 z 8 a -6 + 3 z 8 + 3 a z 7 - z 7 a -1 -17 z 7 a -3 -12 z 7 a -5 + z 7 a -7 + 3 a 2 z 6 -16 z 6 a -2 -22 z 6 a -4 -14 z 6 a -6 -5 z 6 + 2 a 3 z 5 -2 a z 5 -7 z 5 a -1 + 12 z 5 a -3 + 11 z 5 a -5 -4 z 5 a -7 + a 4 z 4 -3 a 2 z 4 + 29 z 4 a -2 + 35 z 4 a -4 + 18 z 4 a -6 + 8 z 4 -2 a 3 z 3 + 14 z 3 a -1 + 8 z 3 a -3 - z 3 a -5 + 3 z 3 a -7 -2 a 4 z 2 + a 2 z 2 -28 z 2 a -2 -23 z 2 a -4 -7 z 2 a -6 -9 z 2 -11 z a -1 -11 z a -3 + a 4 + 13 a -2 + 8 a -4 + 5 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

0 is the signature of L11a451. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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