L10a56

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 4 v 2 u 3 -8 v u 3 + 4 u 3 -6 v 2 u 2 + 13 v u 2 -6 u 2 + 4 v 2 u -8 v u + 4 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-11 q^{3/2}+16 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{21}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 3 a z 5 - z 5 a -1 + a 5 z 3 -3 a 3 z 3 + 3 a z 3 - z 3 a -1 - a z + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$-5 a 3 z 9 -5 a z 9 -12 a 4 z 8 -24 a 2 z 8 -12 z 8 -11 a 5 z 7 -14 a 3 z 7 -14 a z 7 -11 z 7 a -1 -5 a 6 z 6 + 16 a 4 z 6 + 42 a 2 z 6 -5 z 6 a -2 + 16 z 6 - a 7 z 5 + 16 a 5 z 5 + 39 a 3 z 5 + 39 a z 5 + 16 z 5 a -1 - z 5 a -3 + 4 a 6 z 4 -4 a 4 z 4 -16 a 2 z 4 + 4 z 4 a -2 -4 z 4 -6 a 5 z 3 -18 a 3 z 3 -18 a z 3 -6 z 3 a -1 - a 3 z - a z - a 2 + a 3 z -1 + a z -1$ (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 L10a56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$