L11a34

(Knotscape image)

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

Visit L11a34's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,11,19,12 X_16,7,17,8 X_8,17,9,18 X_22,15,5,16 X_12,21,13,22 X_20,13,21,14 X_14,19,15,20 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -7, 8, -9, 6, -4, 5, -3, 9, -8, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 5 -4 v u 4 + 5 u 4 + 8 v u 3 -8 u 3 -8 v u 2 + 8 u 2 + 5 v u -4 u -2 v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{12}{q^{11/2}}-\frac{16}{q^{13/2}}+\frac{17}{q^{15/2}}-\frac{18}{q^{17/2}}+\frac{14}{q^{19/2}}-\frac{10}{q^{21/2}}+\frac{6}{q^{23/2}}-\frac{2}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 -2 a 13 z -1 + 3 z 3 a 11 + 7 z a 11 + 4 a 11 z -1 -2 z 5 a 9 -4 z 3 a 9 -2 z a 9 - a 9 z -1 -3 z 5 a 7 -8 z 3 a 7 -5 z a 7 - a 7 z -1 - z 5 a 5 -2 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 4 z 4 a 16 -5 z 2 a 16 + 2 a 16 -2 z 7 a 15 + 5 z 5 a 15 -2 z 3 a 15 - z a 15 -3 z 8 a 14 + 6 z 6 a 14 -2 z 4 a 14 - z 2 a 14 + a 14 -3 z 9 a 13 + 5 z 7 a 13 -6 z 5 a 13 + 11 z 3 a 13 -9 z a 13 + 2 a 13 z -1 - z 10 a 12 -7 z 8 a 12 + 23 z 6 a 12 -32 z 4 a 12 + 24 z 2 a 12 -6 a 12 -7 z 9 a 11 + 13 z 7 a 11 -17 z 5 a 11 + 22 z 3 a 11 -16 z a 11 + 4 a 11 z -1 - z 10 a 10 -10 z 8 a 10 + 25 z 6 a 10 -28 z 4 a 10 + 18 z 2 a 10 -5 a 10 -4 z 9 a 9 + 6 z 5 a 9 -3 z 3 a 9 - z a 9 + a 9 z -1 -6 z 8 a 8 + 6 z 6 a 8 + 2 z 4 a 8 -3 z 2 a 8 + a 8 -6 z 7 a 7 + 11 z 5 a 7 -10 z 3 a 7 + 6 z a 7 - a 7 z -1 -3 z 6 a 6 + 4 z 4 a 6 - z 2 a 6 - z 5 a 5 + 2 z 3 a 5 - z a 5$ (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 L11a34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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