L11n371

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 4 + v w u 4 - v 2 u 3 + v u 3 + 2 v 2 w u 3 -4 v w u 3 + w u 3 + 2 v 2 u 2 -4 v u 2 - v 2 w u 2 + 4 v w u 2 -2 w u 2 + u 2 - v 2 u + 4 v u - v w u + w u -2 u - v + 1$ (db)
Jones polynomial	$q 3 -3 q 2 + 7 q -9 + 12 q -1 -12 q -2 + 12 q -3 -8 q -4 + 6 q -5 -2 q -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 2 z 8 + a 4 z 6 -6 a 2 z 6 + z 6 + 5 a 4 z 4 -14 a 2 z 4 + 4 z 4 - a 6 z 2 + 10 a 4 z 2 -17 a 2 z 2 + 6 z 2 -2 a 6 + 8 a 4 -11 a 2 + 5 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$2 a 3 z 9 + 2 a z 9 + 6 a 4 z 8 + 10 a 2 z 8 + 4 z 8 + 5 a 5 z 7 + 5 a 3 z 7 + 3 a z 7 + 3 z 7 a -1 + a 6 z 6 -18 a 4 z 6 -29 a 2 z 6 + z 6 a -2 -9 z 6 -11 a 5 z 5 -24 a 3 z 5 -21 a z 5 -8 z 5 a -1 + 6 a 6 z 4 + 33 a 4 z 4 + 33 a 2 z 4 -3 z 4 a -2 + 3 z 4 + 3 a 7 z 3 + 17 a 5 z 3 + 30 a 3 z 3 + 21 a z 3 + 5 z 3 a -1 -8 a 6 z 2 -28 a 4 z 2 -26 a 2 z 2 + 3 z 2 a -2 -3 z 2 -2 a 7 z -7 a 5 z -14 a 3 z -10 a z - z a -1 + 2 a 6 + 10 a 4 + 12 a 2 - a -2 + 4 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11n371. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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