L11n428

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n428 at Knotilus!
	[edit L11n428 Quick Notes]

[edit L11n428 Further Notes and Views]

Link Presentations

[edit Notes on L11n428's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_9,20,10,21 X_5,15,6,14 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_3,10,4,11 X_18,11,19,12 X_15,1,16,6 X_19,4,20,5 X_2,21,3,22
Gauss code	{1, -11, -7, 10, -3, 9}, {5, -1, -2, 7, 8, -4}, {4, 3, -9, -5, 6, -8, -10, 2, 11, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)(uv-1)^{2}}{uv{\sqrt {w}}}}$ (db)
Jones polynomial	$-q^{4}+q^{3}+q+1+q^{-2}+q^{-4}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{4}z^{-2}+2a^{4}-a^{2}z^{4}-z^{4}a^{-2}-6a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}-7a^{2}-2a^{-2}+z^{6}+6z^{4}+10z^{2}+z^{-2}+7$ (db)
Kauffman polynomial	$a^{4}z^{4}-4a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+z^{7}a^{-3}-6z^{5}a^{-3}+a^{3}z^{3}+9z^{3}a^{-3}-3a^{3}z-2za^{-3}+2a^{3}z^{-1}+z^{8}a^{-2}-2a^{2}z^{6}-7z^{6}a^{-2}+13a^{2}z^{4}+15z^{4}a^{-2}-24a^{2}z^{2}-12z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+4a^{-2}+z^{7}a^{-1}-az^{5}-7z^{5}a^{-1}+5az^{3}+13z^{3}a^{-1}-7az-6za^{-1}+2az^{-1}+z^{8}-9z^{6}+27z^{4}-32z^{2}-z^{-2}+13$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

9

1

-1

7

0

5

1

3

1

2

1

2

-1

1

3

1

-3

1

-5

1

3

1

-7

1

-9

1

Integral Khovanov Homology

(db, data source)

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