L11n297

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_15,20,16,21 X_14,8,15,7 X_21,10,22,5 X_18,11,19,12 X_9,17,10,16 X_22,17,11,18 X_8,19,9,20 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -7, 5}, {6, -2, 11, -4, -3, 7, 8, -6, 9, 3, -5, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v 2 w u 2 + v w u 2 - w u 2 -2 v 2 u + 2 v u + v 2 w u -2 v w u + 2 w u - u + v 2 - v - w + 1$ (db)
Jones polynomial	$q 2 -2 q + 5-4 q -1 + 7 q -2 -5 q -3 + 5 q -4 -4 q -5 + 2 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 z -2 -2 a 6 + 2 z 4 a 4 + 7 z 2 a 4 + 4 a 4 z -2 + 8 a 4 - z 6 a 2 -5 z 4 a 2 -10 z 2 a 2 -5 a 2 z -2 -10 a 2 + z 4 + 3 z 2 + 2 z -2 + 4$ (db)
Kauffman polynomial	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 5 a 4 z 8 + 3 a 2 z 8 + a 7 z 7 + a 3 z 7 + 2 a z 7 -9 a 6 z 6 -22 a 4 z 6 -13 a 2 z 6 -5 a 7 z 5 -16 a 5 z 5 -17 a 3 z 5 -6 a z 5 + 11 a 6 z 4 + 30 a 4 z 4 + 24 a 2 z 4 + 5 z 4 + 8 a 7 z 3 + 29 a 5 z 3 + 29 a 3 z 3 + 10 a z 3 + 2 z 3 a -1 -5 a 6 z 2 -22 a 4 z 2 -27 a 2 z 2 + z 2 a -2 -9 z 2 -5 a 7 z -18 a 5 z -24 a 3 z -11 a z + 3 a 6 + 12 a 4 + 15 a 2 + 7 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 L11n297. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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