L11n393

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n393 at Knotilus!
	[edit L11n393 Quick Notes]

[edit L11n393 Further Notes and Views]

Link Presentations

[edit Notes on L11n393's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_12,6,13,5 X₈₄₉₃ X_13,22,14,19 X_9,20,10,21 X_19,10,20,11 X_21,14,22,15 X_18,12,5,11 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {4, -1, 2, -5, -7, 8, 10, -4, -6, 9, 11, -2, 3, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}\left(t(3)^{3}+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{-7}+3q^{-6}-4q^{-5}+7q^{-4}+q^{3}-5q^{-3}-q^{2}+6q^{-2}-q-4q^{-1}+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)+a^{6}z^{-2}-a^{6}+2a^{4}z^{4}+5a^{4}z^{2}-2a^{4}z^{-2}+a^{4}-a^{2}z^{6}-4a^{2}z^{4}-4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}-z^{2}-1$ (db)
Kauffman polynomial	$a^{7}z^{7}-4a^{7}z^{5}+4a^{7}z^{3}-a^{7}z+3a^{6}z^{8}-14a^{6}z^{6}+19a^{6}z^{4}-10a^{6}z^{2}+a^{6}z^{-2}+2a^{5}z^{9}-5a^{5}z^{7}-7a^{5}z^{5}+14a^{5}z^{3}-3a^{5}z-2a^{5}z^{-1}+8a^{4}z^{8}-39a^{4}z^{6}+55a^{4}z^{4}-31a^{4}z^{2}+2a^{4}z^{-2}+4a^{4}+2a^{3}z^{9}-4a^{3}z^{7}-15a^{3}z^{5}+30a^{3}z^{3}-11a^{3}z-2a^{3}z^{-1}+5a^{2}z^{8}-27a^{2}z^{6}+z^{6}a^{-2}+41a^{2}z^{4}-5z^{4}a^{-2}-23a^{2}z^{2}+4z^{2}a^{-2}+a^{2}z^{-2}+5a^{2}-a^{-2}+3az^{7}+z^{7}a^{-1}-19az^{5}-7z^{5}a^{-1}+30az^{3}+10z^{3}a^{-1}-13az-4za^{-1}-z^{6}+2z^{2}+1$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

0

3

2

1

-2

1

3

1

2

-1

4

1

-1

-3

3

2

-5

3

4

1

-7

4

3

1

2

-9

2

5

3

-11

1

2

-1

-13

2

-15

1

-1

Integral Khovanov Homology

(db, data source)

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