L11a496

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 5 - v w u 5 + w u 5 - u 5 -2 v u 4 + 2 v w u 4 -2 w u 4 + 2 u 4 + 2 v u 3 -2 v w u 3 + 2 w u 3 -2 u 3 -2 v u 2 + 2 v w u 2 -2 w u 2 + 2 u 2 + 2 v u -2 v w u + 2 w u -2 u - v + v w - w + 1$ (db)
Jones polynomial	$- q 9 + 3 q 8 -6 q 7 + 8 q 6 -11 q 5 + 13 q 4 -11 q 3 + 11 q 2 -7 q + 6-2 q -1 + q -2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 -2 z 6 a -2 + 6 z 6 a -4 - z 6 a -6 -10 z 4 a -2 + 14 z 4 a -4 -4 z 4 a -6 + z 4 -17 z 2 a -2 + 18 z 2 a -4 -5 z 2 a -6 + 4 z 2 -14 a -2 + 13 a -4 -4 a -6 + 5-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 7 z 9 a -3 + 5 z 9 a -5 + 3 z 8 a -2 + 11 z 8 a -4 + 9 z 8 a -6 + z 8 -9 z 7 a -1 -25 z 7 a -3 -7 z 7 a -5 + 9 z 7 a -7 -34 z 6 a -2 -60 z 6 a -4 -24 z 6 a -6 + 8 z 6 a -8 -6 z 6 + 10 z 5 a -1 + 10 z 5 a -3 -22 z 5 a -5 -16 z 5 a -7 + 6 z 5 a -9 + 70 z 4 a -2 + 88 z 4 a -4 + 19 z 4 a -6 -10 z 4 a -8 + 3 z 4 a -10 + 14 z 4 + 4 z 3 a -1 + 34 z 3 a -3 + 41 z 3 a -5 + 5 z 3 a -7 -5 z 3 a -9 + z 3 a -11 -57 z 2 a -2 -57 z 2 a -4 -14 z 2 a -6 + 2 z 2 a -8 -16 z 2 -12 z a -1 -31 z a -3 -25 z a -5 -4 z a -7 + 2 z a -9 + 23 a -2 + 22 a -4 + 7 a -6 + 9 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 =

4 is the signature of L11a496. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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