L11n262

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n262 at Knotilus!
	[edit L11n262 Quick Notes]

[edit L11n262 Further Notes and Views]

Link Presentations

[edit Notes on L11n262's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,18,12,19 X_19,22,20,9 X_15,20,16,21 X_21,16,22,17 X_17,12,18,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, -7, 8, -9, 5, -6, 7, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{3}-uvw+uv+uw^{3}-3uw^{2}+3uw-u+vw^{3}-3vw^{2}+3vw-v-w^{3}+w^{2}-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-13}+2q^{-12}-5q^{-11}+6q^{-10}-6q^{-9}+7q^{-8}-4q^{-7}+5q^{-6}-q^{-5}+q^{-3}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+3a^{12}z^{-2}+4a^{12}-5z^{2}a^{10}-2a^{10}z^{-2}-7a^{10}+z^{4}a^{8}-a^{8}z^{-2}-a^{8}+z^{6}a^{6}+6z^{4}a^{6}+8z^{2}a^{6}+a^{6}z^{-2}+4a^{6}$ (db)
Kauffman polynomial	$a^{15}z^{7}-5a^{15}z^{5}+9a^{15}z^{3}-7a^{15}z+2a^{15}z^{-1}+2a^{14}z^{8}-8a^{14}z^{6}+8a^{14}z^{4}-2a^{14}z^{2}-a^{14}z^{-2}+a^{14}+a^{13}z^{9}+2a^{13}z^{7}-24a^{13}z^{5}+39a^{13}z^{3}-27a^{13}z+8a^{13}z^{-1}+6a^{12}z^{8}-24a^{12}z^{6}+24a^{12}z^{4}-10a^{12}z^{2}-3a^{12}z^{-2}+5a^{12}+a^{11}z^{9}+4a^{11}z^{7}-33a^{11}z^{5}+52a^{11}z^{3}-34a^{11}z+10a^{11}z^{-1}+4a^{10}z^{8}-18a^{10}z^{6}+24a^{10}z^{4}-13a^{10}z^{2}-2a^{10}z^{-2}+4a^{10}+3a^{9}z^{7}-15a^{9}z^{5}+22a^{9}z^{3}-10a^{9}z+2a^{9}z^{-1}-a^{8}z^{6}+2a^{8}z^{4}+3a^{8}z^{2}+a^{8}z^{-2}-3a^{8}-a^{7}z^{5}+4a^{7}z-2a^{7}z^{-1}+a^{6}z^{6}-6a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-4a^{6}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

1

-9

2

1

-1

-11

4

-13

4

6

1

-15

4

1

3

-17

2

4

1

-19

4

0

-21

1

2

1

-23

1

4

-3

-25

1

-27

1

-1

Integral Khovanov Homology

(db, data source)

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