L11a215

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_22,12,7,11 X_20,16,21,15 X_18,10,19,9 X_10,20,11,19 X_14,22,15,21 X_16,6,17,5 X₂₇₃₈ X_4,14,5,13 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -6, 3, -2, 10, -7, 4, -8, 11, -5, 6, -4, 7, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 3 v u 4 - u 4 + 4 v 2 u 3 -6 v u 3 + 4 u 3 -5 v 2 u 2 + 7 v u 2 -5 u 2 + 4 v 2 u -6 v u + 4 u - v 2 + 3 v -2$ (db)
Jones polynomial	$q^{23/2}-3 q^{21/2}+7 q^{19/2}-12 q^{17/2}+16 q^{15/2}-18 q^{13/2}+18 q^{11/2}-16 q^{9/2}+11 q^{7/2}-8 q^{5/2}+3 q^{3/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 - z 7 a -7 + z 5 a -3 -3 z 5 a -5 -4 z 5 a -7 + z 5 a -9 + 3 z 3 a -3 - z 3 a -5 -7 z 3 a -7 + 3 z 3 a -9 + 3 z a -3 + 2 z a -5 -7 z a -7 + 3 z a -9 + a -3 z -1 -2 a -7 z -1 + a -9 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -6 -2 z 10 a -8 -4 z 9 a -5 -11 z 9 a -7 -7 z 9 a -9 -3 z 8 a -4 -4 z 8 a -6 -11 z 8 a -8 -10 z 8 a -10 - z 7 a -3 + 10 z 7 a -5 + 30 z 7 a -7 + 10 z 7 a -9 -9 z 7 a -11 + 10 z 6 a -4 + 29 z 6 a -6 + 45 z 6 a -8 + 20 z 6 a -10 -6 z 6 a -12 + 4 z 5 a -3 - z 5 a -5 -22 z 5 a -7 + 14 z 5 a -11 -3 z 5 a -13 -9 z 4 a -4 -33 z 4 a -6 -50 z 4 a -8 -19 z 4 a -10 + 6 z 4 a -12 - z 4 a -14 -6 z 3 a -3 -6 z 3 a -5 + 11 z 3 a -7 -2 z 3 a -9 -11 z 3 a -11 + 2 z 3 a -13 + z 2 a -4 + 15 z 2 a -6 + 27 z 2 a -8 + 9 z 2 a -10 -3 z 2 a -12 + z 2 a -14 + 4 z a -3 -7 z a -7 - z a -9 + 2 z a -11 + a -4 -3 a -6 -5 a -8 -2 a -10 - a -3 z -1 + 2 a -7 z -1 + a -9 z -1$ (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 =

5 is the signature of L11a215. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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