L11a498

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 4 v u 4 -2 v w u 4 -2 u 4 -7 v u 3 + 6 v w u 3 -4 w u 3 + 6 u 3 + 4 v u 2 -6 v w u 2 + 7 w u 2 -6 u 2 + 2 v w u -4 w u + 2 u + w$ (db)
Jones polynomial	$q 4 -4 q 3 + 10 q 2 -14 q + 19-20 q -1 + 21 q -2 -16 q -3 + 12 q -4 -7 q -5 + 3 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 z -2 -2 a 6 + 3 z 4 a 4 + 9 z 2 a 4 + 4 a 4 z -2 + 10 a 4 -2 z 6 a 2 -8 z 4 a 2 -15 z 2 a 2 -5 a 2 z -2 -13 a 2 - z 6 - z 4 + 2 z 2 + 2 z -2 + 4 + z 4 a -2 + z 2 a -2 + a -2$ (db)
Kauffman polynomial	$a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 9 a 3 z 9 + 6 a z 9 + 3 a 6 z 8 + 12 a 4 z 8 + 22 a 2 z 8 + 13 z 8 + a 7 z 7 -3 a 5 z 7 -8 a 3 z 7 + 10 a z 7 + 14 z 7 a -1 -11 a 6 z 6 -49 a 4 z 6 -66 a 2 z 6 + 10 z 6 a -2 -18 z 6 -4 a 7 z 5 -17 a 5 z 5 -39 a 3 z 5 -48 a z 5 -18 z 5 a -1 + 4 z 5 a -3 + 14 a 6 z 4 + 60 a 4 z 4 + 66 a 2 z 4 -10 z 4 a -2 + z 4 a -4 + 9 z 4 + 6 a 7 z 3 + 34 a 5 z 3 + 66 a 3 z 3 + 44 a z 3 + 6 z 3 a -1 -8 a 6 z 2 -35 a 4 z 2 -43 a 2 z 2 + 6 z 2 a -2 -10 z 2 -4 a 7 z -21 a 5 z -39 a 3 z -22 a z + 3 a 6 + 16 a 4 + 21 a 2 -2 a -2 + 7 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 =

0 is the signature of L11a498. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}$	${\mathbb Z}$