L11n37

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n37 at Knotilus!
	[edit L11n37 Quick Notes]

[edit L11n37 Further Notes and Views]

Link Presentations

[edit Notes on L11n37's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_11,14,12,15 X₈₄₉₃ X_5,13,6,12 X_13,5,14,22 X_15,19,16,18 X_9,17,10,16 X_17,11,18,10 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -9, 10, -4, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+6q^{11/2}-8q^{9/2}+9q^{7/2}-10q^{5/2}+8q^{3/2}-7{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2z^{5}a^{-3}-3z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+az-5za^{-1}+9za^{-3}-6za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+3a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-3z^{8}a^{-2}-6z^{8}a^{-4}-3z^{8}a^{-6}-2z^{7}a^{-1}-5z^{7}a^{-3}-6z^{7}a^{-5}-3z^{7}a^{-7}+9z^{6}a^{-2}+15z^{6}a^{-4}+5z^{6}a^{-6}-z^{6}a^{-8}+6z^{5}a^{-1}+24z^{5}a^{-3}+27z^{5}a^{-5}+9z^{5}a^{-7}-15z^{4}a^{-2}-9z^{4}a^{-4}+6z^{4}a^{-6}+3z^{4}a^{-8}-3z^{4}-az^{3}-16z^{3}a^{-1}-36z^{3}a^{-3}-28z^{3}a^{-5}-7z^{3}a^{-7}+7z^{2}a^{-2}+z^{2}a^{-4}-7z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}+2az+11za^{-1}+20za^{-3}+14za^{-5}+3za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

4

1

-3

10

4

2

8

5

4

-1

6

5

4

1

4

3

5

2

4

5

-1

0

1

5

4

-2

2

-2

-4

1

Integral Khovanov Homology

(db, data source)

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