L11a175

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_20,12,21,11 X_14,6,15,5 X_4,21,5,22 X_16,7,17,8 X_22,15,7,16 X_18,14,19,13 X_6,18,1,17 X_12,20,13,19
Gauss code	{1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -11, 9, -5, 8, -7, 10, -9, 11, -4, 6, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 4 v 2 u 3 -5 v u 3 + 2 u 3 -4 v 2 u 2 + 5 v u 2 -4 u 2 + 2 v 2 u -5 v u + 4 u + 2 v -2$ (db)
Jones polynomial	$q^{7/2}-3 q^{5/2}+5 q^{3/2}-9 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 3 z 7 - a z 7 + a 5 z 5 -4 a 3 z 5 -4 a z 5 + z 5 a -1 + 3 a 5 z 3 -5 a 3 z 3 -4 a z 3 + 3 z 3 a -1 + 2 a 5 z -4 a 3 z + z a -1 + a 5 z -1 -2 a 3 z -1 + 2 a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -6 a 3 z 9 -9 a z 9 -3 z 9 a -1 -10 a 4 z 8 -4 a 2 z 8 - z 8 a -2 + 5 z 8 -11 a 5 z 7 + 11 a 3 z 7 + 38 a z 7 + 16 z 7 a -1 -9 a 6 z 6 + 25 a 4 z 6 + 36 a 2 z 6 + 5 z 6 a -2 + 7 z 6 -6 a 7 z 5 + 23 a 5 z 5 + 12 a 3 z 5 -45 a z 5 -28 z 5 a -1 -3 a 8 z 4 + 12 a 6 z 4 -14 a 4 z 4 -43 a 2 z 4 -8 z 4 a -2 -22 z 4 - a 9 z 3 + 4 a 7 z 3 -12 a 5 z 3 -22 a 3 z 3 + 12 a z 3 + 17 z 3 a -1 -3 a 6 z 2 + 4 a 4 z 2 + 13 a 2 z 2 + 4 z 2 a -2 + 10 z 2 - a 7 z + 4 a 5 z + 10 a 3 z + 4 a z - z a -1 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 =

-3 is the signature of L11a175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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