L11n313

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_9,20,10,21 X_7,16,8,17 X_13,18,14,19 X_19,14,20,15 X_15,22,16,11 X_17,10,18,5 X_21,8,22,9 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 9, -3, 8}, {11, -2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 4 - v 2 u 3 + v u 3 - v 2 w u 3 + v w u 3 + w u 3 + 2 v 2 u 2 - v u 2 + v w u 2 -2 w u 2 - v 2 u - v u - v w u + w u + u + v$ (db)
Jones polynomial	$q -3 - q -4 + q -5 + q -6 + 3 q -8 -2 q -9 + 3 q -10 -3 q -11 + 2 q -12 - q -13$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 14 z -2 + 4 a 12 z -2 + 5 a 12 - z 4 a 10 -9 z 2 a 10 -5 a 10 z -2 -13 a 10 + z 6 a 8 + 6 z 4 a 8 + 9 z 2 a 8 + 2 a 8 z -2 + 7 a 8 + z 6 a 6 + 5 z 4 a 6 + 5 z 2 a 6 + a 6$ (db)
Kauffman polynomial	$z 7 a 15 -5 z 5 a 15 + 7 z 3 a 15 -4 z a 15 + a 15 z -1 + 2 z 8 a 14 -10 z 6 a 14 + 13 z 4 a 14 -7 z 2 a 14 - a 14 z -2 + 3 a 14 + z 9 a 13 -2 z 7 a 13 -11 z 5 a 13 + 26 z 3 a 13 -19 z a 13 + 5 a 13 z -1 + 4 z 8 a 12 -23 z 6 a 12 + 38 z 4 a 12 -29 z 2 a 12 -4 a 12 z -2 + 15 a 12 + z 9 a 11 -2 z 7 a 11 -16 z 5 a 11 + 42 z 3 a 11 -33 z a 11 + 9 a 11 z -1 + 3 z 8 a 10 -21 z 6 a 10 + 44 z 4 a 10 -42 z 2 a 10 -5 a 10 z -2 + 20 a 10 + 2 z 7 a 9 -15 z 5 a 9 + 27 z 3 a 9 -18 z a 9 + 5 a 9 z -1 + z 8 a 8 -7 z 6 a 8 + 14 z 4 a 8 -15 z 2 a 8 -2 a 8 z -2 + 8 a 8 + z 7 a 7 -5 z 5 a 7 + 4 z 3 a 7 + z 6 a 6 -5 z 4 a 6 + 5 z 2 a 6 - a 6$ (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 L11n313. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -7$	$i = -5$	$i = -3$
$r = -11$		${\mathbb Z}$
$r = -10$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -7$	${\mathbb Z}$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{2}$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$