L11n298

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_13,19,14,18 X_17,11,18,22 X_7,17,8,16 X_21,8,22,9 X_9,14,10,15 X_15,20,16,21 X_19,5,20,10 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -5, 6, -7, 9}, {-11, 2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - u 4 -3 v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 + 3 u 3 - v 2 u 2 + 5 v u 2 + 3 v 2 w u 2 -5 v w u 2 + w u 2 -3 u 2 + v 2 u -2 v u -3 v 2 w u + 3 v w u + u + v 2 w - v w$ (db)
Jones polynomial	$q 6 -4 q 5 + 8 q 4 -11 q 3 + 14 q 2 -14 q + 14-9 q -1 + 7 q -2 -2 q -3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 -3 z 4 a -2 + z 4 a -4 + 3 z 4 -2 a 2 z 2 -5 z 2 a -2 + z 2 a -4 + 5 z 2 -2 a -2 + a -4 + 1 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$3 z 9 a -1 + 3 z 9 a -3 + 14 z 8 a -2 + 6 z 8 a -4 + 8 z 8 + 6 a z 7 + 5 z 7 a -1 + 3 z 7 a -3 + 4 z 7 a -5 + a 2 z 6 -40 z 6 a -2 -15 z 6 a -4 + z 6 a -6 -23 z 6 -11 a z 5 -28 z 5 a -1 -27 z 5 a -3 -10 z 5 a -5 + 8 a 2 z 4 + 39 z 4 a -2 + 8 z 4 a -4 -2 z 4 a -6 + 37 z 4 + 3 a 3 z 3 + 14 a z 3 + 30 z 3 a -1 + 25 z 3 a -3 + 6 z 3 a -5 -9 a 2 z 2 -20 z 2 a -2 -3 z 2 a -4 + z 2 a -6 -25 z 2 - a 3 z -3 a z -7 z a -1 -7 z a -3 -2 z a -5 + 4 a -2 + 2 a -4 + 3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 L11n298. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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