L11n370

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,8,15,7 X_15,20,16,21 X_11,19,12,18 X_17,13,18,12 X_19,22,20,17 X_21,16,22,5 X_8,14,9,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7}, {10, -1, 3, -9, 11, -2, -5, 6, 9, -3, -4, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -3 v u 2 + 3 v w u 2 -3 w u 2 + 3 u 2 + 3 v u -3 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$q 3 -3 q 2 + 6 q -9 + 11 q -1 -10 q -2 + 11 q -3 -7 q -4 + 5 q -5 - q -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 6 z -2 - z 4 a 4 - z 2 a 4 -2 a 4 z -2 -2 a 4 + z 6 a 2 + 3 z 4 a 2 + 4 z 2 a 2 + a 2 z -2 + 3 a 2 -2 z 4 -4 z 2 -2 + z 2 a -2 + a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 3 a 4 z 8 + 6 a 2 z 8 + 3 z 8 + 2 a 5 z 7 + 6 a 3 z 7 + 7 a z 7 + 3 z 7 a -1 -4 a 4 z 6 -9 a 2 z 6 + z 6 a -2 -4 z 6 -16 a 3 z 5 -25 a z 5 -9 z 5 a -1 + 5 a 6 z 4 + 6 a 4 z 4 -3 a 2 z 4 -3 z 4 a -2 -7 z 4 + a 7 z 3 + 2 a 5 z 3 + 12 a 3 z 3 + 19 a z 3 + 8 z 3 a -1 - a 6 z 2 -3 a 4 z 2 + 3 a 2 z 2 + 3 z 2 a -2 + 8 z 2 + 2 a 5 z -2 a 3 z -6 a z -2 z a -1 -2 a 6 -2 a 4 - a 2 - a -2 -1-2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 =

-2 is the signature of L11n370. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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