L11a541

From Knot Atlas

Jump to: navigation, search

L11a540

L11a542

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 L11a541's page at Knotilus.

Visit L11a541's page at the original Knot Atlas.

[edit L11a541 Quick Notes]

[edit L11a541 Further Notes and Views]

[edit] Link Presentations

[edit Notes on L11a541's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 v w u 3 - w u 3 + v x u 3 - v w x u 3 + w x u 3 - x u 3 + u 3 + 5 v u 2 -5 v w u 2 + 3 w u 2 -3 v x u 2 + 3 v w x u 2 -4 w x u 2 + 4 x u 2 -3 u 2 -4 v u + 4 v w u -3 w u + 3 v x u -3 v w x u + 5 w x u -5 x u + 3 u + v - v w + w - v x + v w x -2 w x + 2 x -1$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-13 q^{3/2}+18 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{25}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{20}{q^{7/2}}-\frac{15}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 6 z a 5 + 5 a 5 z -1 + a 5 z -3 -3 z 5 a 3 -8 z 3 a 3 -12 z a 3 -10 a 3 z -1 -3 a 3 z -3 + z 7 a + 3 z 5 a + 6 z 3 a + 9 z a + 9 a z -1 + 3 a z -3 - z 5 a -1 - z 3 a -1 -2 z a -1 -3 a -1 z -1 - a -1 z -3$ (db)
Kauffman polynomial	$-2 a 4 z 10 -2 a 2 z 10 -4 a 5 z 9 -13 a 3 z 9 -9 a z 9 -4 a 6 z 8 -12 a 4 z 8 -24 a 2 z 8 -16 z 8 -3 a 7 z 7 -4 a 5 z 7 + 5 a 3 z 7 -7 a z 7 -13 z 7 a -1 - a 8 z 6 + 3 a 6 z 6 + 24 a 4 z 6 + 48 a 2 z 6 -5 z 6 a -2 + 23 z 6 + 8 a 7 z 5 + 28 a 5 z 5 + 44 a 3 z 5 + 43 a z 5 + 18 z 5 a -1 - z 5 a -3 + 3 a 8 z 4 + 9 a 6 z 4 + 2 a 4 z 4 -11 a 2 z 4 + 2 z 4 a -2 -5 z 4 -9 a 7 z 3 -43 a 5 z 3 -68 a 3 z 3 -43 a z 3 -9 z 3 a -1 -3 a 8 z 2 -14 a 6 z 2 -30 a 4 z 2 -24 a 2 z 2 -5 z 2 + 6 a 7 z + 31 a 5 z + 48 a 3 z + 30 a z + 7 z a -1 + a 8 + 6 a 6 + 18 a 4 + 21 a 2 + 9-2 a 7 z -1 -11 a 5 z -1 -18 a 3 z -1 -14 a z -1 -5 a -1 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L11a541. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a541/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = 0$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{16}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$