L11n274

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_9,18,10,19 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, {-8, 2, -4, -5, 10, 9, -7, 6, -11, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 4 + 2 v 2 w u 3 - v w u 3 - v u 2 -2 v 2 w u 2 + v w u 2 + 2 u 2 + v u -2 u + 1$ (db)
Jones polynomial	$- q + 2-3 q -1 + 4 q -2 -4 q -3 + 5 q -4 -3 q -5 + 4 q -6 - q -7 + q -8$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 2 a 8 + 2 a 8 z -2 + 2 a 8 - z 6 a 6 -6 z 4 a 6 -12 z 2 a 6 -5 a 6 z -2 -12 a 6 + z 8 a 4 + 7 z 6 a 4 + 18 z 4 a 4 + 23 z 2 a 4 + 4 a 4 z -2 + 14 a 4 - z 6 a 2 -5 z 4 a 2 -7 z 2 a 2 - a 2 z -2 -4 a 2$ (db)
Kauffman polynomial	$z 2 a 10 - a 10 + z 3 a 9 + 3 z 4 a 8 -8 z 2 a 8 -2 a 8 z -2 + 8 a 8 + 2 z 7 a 7 -10 z 5 a 7 + 20 z 3 a 7 -18 z a 7 + 5 a 7 z -1 + 3 z 8 a 6 -16 z 6 a 6 + 31 z 4 a 6 -33 z 2 a 6 -5 a 6 z -2 + 20 a 6 + z 9 a 5 -19 z 5 a 5 + 41 z 3 a 5 -33 z a 5 + 9 a 5 z -1 + 5 z 8 a 4 -26 z 6 a 4 + 42 z 4 a 4 -32 z 2 a 4 -4 a 4 z -2 + 15 a 4 + z 9 a 3 - z 7 a 3 -14 z 5 a 3 + 29 z 3 a 3 -19 z a 3 + 5 a 3 z -1 + 2 z 8 a 2 -10 z 6 a 2 + 14 z 4 a 2 -8 z 2 a 2 - a 2 z -2 + 3 a 2 + z 7 a -5 z 5 a + 7 z 3 a -4 z a + a z -1$ (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 L11n274. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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