L11a546

From Knot Atlas

Jump to: navigation, search

L11a545

L11a547

1 Link Presentations
2 Polynomial invariants
3 Khovanov Homology
4 Computer Talk
5 Modifying This Page

(Knotscape image)

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

Visit L11a546's page at Knotilus.

Visit L11a546's page at the original Knot Atlas.

[edit L11a546 Quick Notes]

[edit L11a546 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a546's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$3 v 2 u 2 -3 v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 - v 2 x u 2 + 2 v x u 2 - v w x u 2 + w x u 2 - x u 2 + u 2 -3 v 2 u + 3 v u + 2 v 2 w u -4 v w u + 2 w u + 2 v 2 x u -4 v x u - v 2 w x u + 3 v w x u -3 w x u + 2 x u - u + v 2 - v - v 2 w + 2 v w - w - v 2 x + 2 v x + v 2 w x -3 v w x + 3 w x - x$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+9 \sqrt{q}-\frac{15}{\sqrt{q}}+\frac{18}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{19}{q^{7/2}}-\frac{18}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 + a 9 z -3 -4 z a 7 -6 a 7 z -1 -3 a 7 z -3 + 6 z 3 a 5 + 12 z a 5 + 9 a 5 z -1 + 3 a 5 z -3 -3 z 5 a 3 -8 z 3 a 3 -10 z a 3 -4 a 3 z -1 - a 3 z -3 - z 5 a + z 3 a + 2 z a + z 3 a -1$ (db)
Kauffman polynomial	$- a 6 z 10 - a 4 z 10 -3 a 7 z 9 -9 a 5 z 9 -6 a 3 z 9 -2 a 8 z 8 -11 a 6 z 8 -22 a 4 z 8 -13 a 2 z 8 - a 9 z 7 + 5 a 7 z 7 + 8 a 5 z 7 -12 a 3 z 7 -14 a z 7 + 5 a 8 z 6 + 40 a 6 z 6 + 61 a 4 z 6 + 17 a 2 z 6 -9 z 6 + 5 a 9 z 5 + 7 a 7 z 5 + 34 a 5 z 5 + 57 a 3 z 5 + 21 a z 5 -4 z 5 a -1 -31 a 6 z 4 -41 a 4 z 4 -2 a 2 z 4 - z 4 a -2 + 7 z 4 -10 a 9 z 3 -22 a 7 z 3 -49 a 5 z 3 -53 a 3 z 3 -15 a z 3 + z 3 a -1 -10 a 8 z 2 -10 a 6 z 2 -2 a 2 z 2 -2 z 2 + 10 a 9 z + 23 a 7 z + 27 a 5 z + 20 a 3 z + 6 a z + 10 a 8 + 19 a 6 + 10 a 4 -5 a 9 z -1 -12 a 7 z -1 -12 a 5 z -1 -5 a 3 z -1 -3 a 8 z -2 -6 a 6 z -2 -3 a 4 z -2 + a 9 z -3 + 3 a 7 z -3 + 3 a 5 z -3 + a 3 z -3$ (db)

[edit] Khovanov Homology

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 L11a546. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a546/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -8$	${\mathbb Z}$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -4$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$