L10n34

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n34 at Knotilus!
	[edit L10n34 Quick Notes]

[edit L10n34 Further Notes and Views]

Link Presentations

[edit Notes on L10n34's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,8,17,7 X_17,20,18,5 X_11,19,12,18 X_19,11,20,10 X_14,10,15,9 X_8,16,9,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 7, 6, -5, -2, 10, -7, 8, -3, -4, 5, -6, 4}

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

Polynomial invariants

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

0

1

2

3

4

5

6

7

χ

16

1

-1

14

1

12

2

1

-1

10

2

1

8

2

0

6

2

0

4

1

2

1

2

0

3

Integral Khovanov Homology

(db, data source)

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