L11n395

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n395 at Knotilus!
	[edit L11n395 Quick Notes]

[edit L11n395 Further Notes and Views]

Link Presentations

[edit Notes on L11n395's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_15,1,16,4 X_9,22,10,19 X₃₈₄₉ X_21,17,22,16 X_11,5,12,18 X_5,21,6,20 X_17,11,18,10 X_19,12,20,13 X_2,14,3,13
Gauss code	{1, -11, -5, 3}, {-10, 8, -6, 4}, {-8, -1, 2, 5, -4, 9, -7, 10, 11, -2, -3, 6, -9, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{5}-4q^{4}-q^{-4}+6q^{3}+3q^{-3}-6q^{2}-4q^{-2}+9q+7q^{-1}-7$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}-a^{2}z^{4}-2z^{4}a^{-2}+4z^{4}-2a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+5z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+5z^{8}a^{-2}+8z^{8}+a^{3}z^{7}-5az^{7}-z^{7}a^{-1}+5z^{7}a^{-3}-14a^{2}z^{6}-18z^{6}a^{-2}+2z^{6}a^{-4}-34z^{6}-4a^{3}z^{5}-5az^{5}-17z^{5}a^{-1}-16z^{5}a^{-3}+19a^{2}z^{4}+22z^{4}a^{-2}-z^{4}a^{-4}+42z^{4}+4a^{3}z^{3}+13az^{3}+22z^{3}a^{-1}+17z^{3}a^{-3}+4z^{3}a^{-5}-8a^{2}z^{2}-13z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-19z^{2}-a^{3}z-4az-6za^{-1}-4za^{-3}-za^{-5}+1-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

11

1

9

3

-3

7

4

2

5

3

0

3

6

4

1

3

1

5

7

2

-1

2

3

1

0

-3

2

5

3

-5

1

2

-1

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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