L11a315

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_22,17,9,18 X_20,9,21,10 X_6,13,7,14 X_14,7,15,8 X_8,15,1,16 X_4,19,5,20 X_18,5,19,6 X_16,21,17,22
Gauss code	{1, -2, 3, -9, 10, -6, 7, -8}, {5, -1, 2, -3, 6, -7, 8, -11, 4, -10, 9, -5, 11, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 -2 v 3 u 4 + 4 v 2 u 4 -2 v u 4 + 2 v 3 u 3 -7 v 2 u 3 + 5 v u 3 - u 3 - v 3 u 2 + 5 v 2 u 2 -7 v u 2 + 2 u 2 -2 v 2 u + 4 v u -2 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{13}{q^{15/2}}+\frac{15}{q^{17/2}}-\frac{15}{q^{19/2}}+\frac{14}{q^{21/2}}-\frac{11}{q^{23/2}}+\frac{6}{q^{25/2}}-\frac{3}{q^{27/2}}+\frac{1}{q^{29/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z 3 a 13 -3 z a 13 - a 13 z -1 + 3 z 5 a 11 + 12 z 3 a 11 + 13 z a 11 + 3 a 11 z -1 -2 z 7 a 9 -10 z 5 a 9 -16 z 3 a 9 -10 z a 9 -2 a 9 z -1 - z 7 a 7 -5 z 5 a 7 -8 z 3 a 7 -4 z a 7$ (db)
Kauffman polynomial	$- z 4 a 18 + z 2 a 18 -3 z 5 a 17 + 3 z 3 a 17 - z a 17 -5 z 6 a 16 + 4 z 4 a 16 - z 2 a 16 -7 z 7 a 15 + 9 z 5 a 15 -5 z 3 a 15 - z a 15 -7 z 8 a 14 + 11 z 6 a 14 -8 z 4 a 14 + 3 z 2 a 14 - a 14 -4 z 9 a 13 + 13 z 5 a 13 -9 z 3 a 13 - z a 13 + a 13 z -1 - z 10 a 12 -10 z 8 a 12 + 35 z 6 a 12 -39 z 4 a 12 + 22 z 2 a 12 -3 a 12 -7 z 9 a 11 + 19 z 7 a 11 -20 z 5 a 11 + 24 z 3 a 11 -16 z a 11 + 3 a 11 z -1 - z 10 a 10 -5 z 8 a 10 + 26 z 6 a 10 -32 z 4 a 10 + 17 z 2 a 10 -3 a 10 -3 z 9 a 9 + 11 z 7 a 9 -16 z 5 a 9 + 17 z 3 a 9 -11 z a 9 + 2 a 9 z -1 -2 z 8 a 8 + 7 z 6 a 8 -6 z 4 a 8 - z 7 a 7 + 5 z 5 a 7 -8 z 3 a 7 + 4 z a 7$ (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 =

-7 is the signature of L11a315. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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