L11n413

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n413 at Knotilus!
	[edit L11n413 Quick Notes]

[edit L11n413 Further Notes and Views]

Link Presentations

[edit Notes on L11n413's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_5,15,6,14 X_10,3,11,4 X_13,5,14,4 X₂₇₃₈ X_6,9,1,10 X_11,18,12,19 X_17,12,18,7 X_20,16,21,15 X_22,20,13,19 X_16,22,17,21
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, 9, -11, -8, 7, 10, -9, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(3)^{3}-t(1)t(2)^{2}t(3)^{2}-t(1)t(3)^{2}-t(1)^{2}t(2)t(3)^{2}+2t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(1)t(2)^{2}t(3)+t(1)t(3)+t(1)^{2}t(2)t(3)-2t(1)t(2)t(3)+t(2)t(3)-t(2)^{2}}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+2q^{4}-3q^{3}+4q^{2}-4q+4-2q^{-1}+2q^{-2}+q^{-3}+q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{2}a^{4}+2a^{4}z^{-2}+3a^{4}-z^{4}a^{2}-6z^{2}a^{2}-5a^{2}z^{-2}-9a^{2}+z^{4}+3z^{2}+4z^{-2}+6+z^{4}a^{-2}+2z^{2}a^{-2}-a^{-2}z^{-2}+a^{-2}-z^{2}a^{-4}-a^{-4}$ (db)
Kauffman polynomial	$a^{4}z^{8}+a^{2}z^{8}+z^{8}a^{-2}+z^{8}+a^{3}z^{7}+2az^{7}+3z^{7}a^{-1}+2z^{7}a^{-3}-8a^{4}z^{6}-10a^{2}z^{6}-2z^{6}a^{-2}+2z^{6}a^{-4}-6z^{6}-9a^{3}z^{5}-16az^{5}-14z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}+21a^{4}z^{4}+31a^{2}z^{4}-z^{4}a^{-2}-6z^{4}a^{-4}+15z^{4}+22a^{3}z^{3}+41az^{3}+26z^{3}a^{-1}+4z^{3}a^{-3}-3z^{3}a^{-5}-23a^{4}z^{2}-39a^{2}z^{2}+3z^{2}a^{-2}+3z^{2}a^{-4}-16z^{2}-19a^{3}z-35az-19za^{-1}-2za^{-3}+za^{-5}+11a^{4}+22a^{2}-a^{-4}+13+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

1

7

2

1

-1

5

2

1

3

2

0

1

3

2

0

-1

2

4

2

-3

1

2

1

0

-5

3

-7

1

-9

1

-11

1

Integral Khovanov Homology

(db, data source)

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