L11n322

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n322 at Knotilus!
	[edit L11n322 Quick Notes]

[edit L11n322 Further Notes and Views]

Link Presentations

[edit Notes on L11n322's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_4,17,1,18 X_19,13,20,22 X_13,10,14,11 X_21,5,22,12 X_11,21,12,20
Gauss code	{1, -4, 3, -7}, {-2, -1, 5, -3, -6, 9, -11, 10}, {-9, 2, 4, -5, 7, 6, -8, 11, -10, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-2q^{3}+5q^{2}-7q+11-10q^{-1}+11q^{-2}-8q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-4a^{2}z^{4}+3z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-2z^{2}a^{-2}+8z^{2}+2a^{4}-7a^{2}-2a^{-2}+7+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-9a^{5}z^{5}+7a^{5}z^{3}-a^{5}z+3a^{4}z^{8}-5a^{4}z^{6}-4a^{4}z^{4}+5a^{4}z^{2}-a^{4}z^{-2}+a^{4}+a^{3}z^{9}+6a^{3}z^{7}-22a^{3}z^{5}+17a^{3}z^{3}+3z^{3}a^{-3}-6a^{3}z-2za^{-3}+2a^{3}z^{-1}+6a^{2}z^{8}-11a^{2}z^{6}+z^{6}a^{-2}+5a^{2}z^{4}+4z^{4}a^{-2}-8a^{2}z^{2}-5z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}+3a^{-2}+az^{9}+6az^{7}+3z^{7}a^{-1}-17az^{5}-4z^{5}a^{-1}+17az^{3}+10z^{3}a^{-1}-10az-7za^{-1}+2az^{-1}+3z^{8}-4z^{6}+10z^{4}-15z^{2}-z^{-2}+9$ (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

χ

7

2

-2

5

3

4

2

-2

1

7

3

4

-1

5

6

1

-3

6

5

1

-5

4

7

3

-7

2

4

-2

-9

1

4

3

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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