L11a15

(Knotscape image)

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

Visit L11a15's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,10,19,9 X₈₄₉₃ X_14,6,15,5 X_22,16,5,15 X_20,18,21,17 X_16,22,17,21 X_10,20,11,19 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4, 10, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(u-1) (v-1) \left(3 v^2-5 v+3\right)}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$-12 q^{9/2}+13 q^{7/2}-14 q^{5/2}+\frac{1}{q^{5/2}}+11 q^{3/2}-\frac{3}{q^{3/2}}-q^{17/2}+4 q^{15/2}-6 q^{13/2}+9 q^{11/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 3 a -7 + a -7 z -1 + z 5 a -5 + z 3 a -5 - z a -5 -2 a -5 z -1 + z 5 a -3 - z a -3 + z 5 a -1 - a z 3 + 2 z 3 a -1 - a z + 3 z a -1 + a -1 z -1$ (db)
Kauffman polynomial	$z 7 a -9 -3 z 5 a -9 + z 3 a -9 + 4 z 8 a -8 -16 z 6 a -8 + 15 z 4 a -8 - z 2 a -8 -2 a -8 + 5 z 9 a -7 -20 z 7 a -7 + 21 z 5 a -7 -6 z 3 a -7 + a -7 z -1 + 2 z 10 a -6 -21 z 6 a -6 + 24 z 4 a -6 + z 2 a -6 -5 a -6 + 9 z 9 a -5 -33 z 7 a -5 + 36 z 5 a -5 -13 z 3 a -5 - z a -5 + 2 a -5 z -1 + 2 z 10 a -4 -11 z 6 a -4 + 9 z 4 a -4 + 2 z 2 a -4 -3 a -4 + 4 z 9 a -3 -8 z 7 a -3 + 9 z 5 a -3 -9 z 3 a -3 + 2 z a -3 + 4 z 8 a -2 -2 z 6 a -2 + a 2 z 4 -4 z 4 a -2 - a 2 z 2 + z 2 a -2 + a -2 + 4 z 7 a -1 + 3 a z 5 -4 a z 3 -7 z 3 a -1 + 2 a z + 5 z a -1 - a -1 z -1 + 4 z 6 -3 z 4$ (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 =

1 is the signature of L11a15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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