L11n304

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n304 at Knotilus!
	[edit L11n304 Quick Notes]

[edit L11n304 Further Notes and Views]

Link Presentations

[edit Notes on L11n304's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,13,21,14 X_22,19,11,20 X_15,5,16,10 X_17,9,18,8 X_7,17,8,16 X_9,19,10,18 X_14,21,15,22 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{2}w^{3}+uv^{2}w^{2}+uvw^{3}-2uvw^{2}+uvw+2uw^{2}-2uw+u-v^{2}w^{4}+2v^{2}w^{3}-2v^{2}w^{2}-vw^{3}+2vw^{2}-vw-w^{2}+w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-4q^{-5}+6q^{-4}-6q^{-3}+2q^{2}+8q^{-2}-3q-6q^{-1}+6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}z^{-2}-3a^{6}+3z^{4}a^{4}+12z^{2}a^{4}+4a^{4}z^{-2}+13a^{4}-2z^{6}a^{2}-11z^{4}a^{2}-21z^{2}a^{2}-5a^{2}z^{-2}-17a^{2}+2z^{4}+7z^{2}+2z^{-2}+7$ (db)
Kauffman polynomial	$a^{7}z^{7}-5a^{7}z^{5}+8a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+2a^{6}z^{8}-9a^{6}z^{6}+12a^{6}z^{4}-7a^{6}z^{2}-a^{6}z^{-2}+4a^{6}+a^{5}z^{9}+a^{5}z^{7}-20a^{5}z^{5}+35a^{5}z^{3}-21a^{5}z+5a^{5}z^{-1}+6a^{4}z^{8}-26a^{4}z^{6}+38a^{4}z^{4}-32a^{4}z^{2}-4a^{4}z^{-2}+17a^{4}+a^{3}z^{9}+4a^{3}z^{7}-28a^{3}z^{5}+44a^{3}z^{3}-33a^{3}z+9a^{3}z^{-1}+4a^{2}z^{8}-15a^{2}z^{6}+26a^{2}z^{4}-32a^{2}z^{2}+3z^{2}a^{-2}-5a^{2}z^{-2}+20a^{2}-2a^{-2}+4az^{7}-12az^{5}+z^{5}a^{-1}+18az^{3}+z^{3}a^{-1}-16az+za^{-1}+5az^{-1}+2z^{6}-4z^{2}-2z^{-2}+6$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

5

2

3

2

-1

1

3

-1

4

0

-3

4

2

-5

2

4

2

-7

4

0

-9

1

3

2

-11

1

3

-2

-13

1

-15

1

-1

Integral Khovanov Homology

(db, data source)

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