L11n97

(Knotscape image)

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

Visit L11n97's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,8,17,7 X_18,10,19,9 X_8,18,9,17 X_19,22,20,5 X_13,20,14,21 X_21,14,22,15 X_10,16,11,15 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 2 v u 4 -2 u 4 -3 v u 3 + 4 u 3 + 4 v u 2 -3 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 - z 5 a -1 -3 a 3 z 3 + 8 a z 3 -3 z 3 a -1 -3 a 3 z + 3 a z -2 z a -1 + a 5 z -1 - a 3 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 - a 4 z 8 -4 a 2 z 8 -3 z 8 + 2 a 3 z 7 -2 a z 7 -4 z 7 a -1 + a 4 z 6 + 10 a 2 z 6 -3 z 6 a -2 + 6 z 6 -4 a 5 z 5 -7 a 3 z 5 + 8 a z 5 + 10 z 5 a -1 - z 5 a -3 - a 6 z 4 -3 a 4 z 4 -13 a 2 z 4 + 7 z 4 a -2 -4 z 4 + 6 a 5 z 3 + 6 a 3 z 3 -9 a z 3 -7 z 3 a -1 + 2 z 3 a -3 + a 6 z 2 + 2 a 4 z 2 + 4 a 2 z 2 -2 z 2 a -2 + z 2 + a 5 z + a z + 2 z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-1 is the signature of L11n97. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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