L11a357

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a357 at Knotilus!
	[edit L11a357 Quick Notes]

[edit L11a357 Further Notes and Views]

Link Presentations

[edit Notes on L11a357's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_18,7,19,8 X_14,3,15,4 X_20,10,21,9 X_22,19,11,20 X_6,15,7,16 X_16,5,17,6 X_4,17,5,18 X_8,22,9,21 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -10, 3, -8, 7, -6, 2, -9, 4, -11}, {10, -1, 11, -3, 6, -7, 8, -2, 5, -4, 9, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u+v-1)(uv-u-v)\left(u^{2}v+uv^{2}-uv+u+v\right)}{u^{2}v^{2}}}$ (db)
Jones polynomial	${\frac {15}{q^{9/2}}}-{\frac {13}{q^{7/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {9}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {14}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-2a^{9}z+a^{7}z^{5}+2a^{7}z^{3}+2a^{7}z+a^{7}z^{-1}+2a^{5}z^{5}+4a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{3}z^{5}-3a^{3}z-az^{3}-az$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+4a^{12}z^{2}+2a^{11}z^{7}-7a^{11}z^{5}+8a^{11}z^{3}-4a^{11}z+2a^{10}z^{8}-3a^{10}z^{6}-3a^{10}z^{4}+4a^{10}z^{2}+2a^{9}z^{9}-3a^{9}z^{7}+a^{9}z^{5}+2a^{9}z^{3}-6a^{9}z+a^{8}z^{10}+a^{8}z^{8}-a^{8}z^{6}-4a^{8}z^{4}+4a^{8}z^{2}+5a^{7}z^{9}-11a^{7}z^{7}+15a^{7}z^{5}-11a^{7}z^{3}+4a^{7}z-a^{7}z^{-1}+a^{6}z^{10}+4a^{6}z^{8}-8a^{6}z^{6}+5a^{6}z^{4}+a^{6}z^{2}+a^{6}+3a^{5}z^{9}-a^{5}z^{7}-3a^{5}z^{5}+2a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}+5a^{4}z^{8}-8a^{4}z^{6}+5a^{4}z^{4}-2a^{4}z^{2}+5a^{3}z^{7}-9a^{3}z^{5}+5a^{3}z^{3}-3a^{3}z+3a^{2}z^{6}-5a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

5

1

4

-4

6

3

-3

-6

7

4

3

-8

8

6

-2

-10

6

7

-1

-12

5

8

3

-14

4

6

-2

-16

1

6

5

-18

1

3

-2

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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