L11n317

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n317 at Knotilus!
	[edit L11n317 Quick Notes]

[edit L11n317 Further Notes and Views]

Link Presentations

[edit Notes on L11n317's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,7,17,8 X_9,20,10,21 X_11,18,12,19 X_19,22,20,11 X_8,15,9,16 X_21,10,22,5 X_14,18,15,17 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, -4, 8}, {-5, -2, 11, -9, 7, -3, 9, 5, -6, 4, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{2}-2uv^{2}w+uv^{2}-3uvw^{2}+5uvw-2uv+uw^{2}-3uw+u-v^{2}w^{2}+3v^{2}w-v^{2}+2vw^{2}-5vw+3v-w^{2}+2w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{2}+4q-7+11q^{-1}-12q^{-2}+14q^{-3}-11q^{-4}+9q^{-5}-5q^{-6}+2q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$2a^{6}z^{2}+a^{6}z^{-2}+2a^{6}-3a^{4}z^{4}-7a^{4}z^{2}-2a^{4}z^{-2}-6a^{4}+a^{2}z^{6}+3a^{2}z^{4}+5a^{2}z^{2}+a^{2}z^{-2}+4a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$3a^{8}z^{4}-4a^{8}z^{2}+a^{8}+a^{7}z^{7}+3a^{7}z^{5}-4a^{7}z^{3}+2a^{6}z^{8}+a^{6}z^{6}-3a^{6}z^{4}+4a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+a^{5}z^{9}+5a^{5}z^{7}-5a^{5}z^{5}-4a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+6a^{4}z^{8}-2a^{4}z^{6}-13a^{4}z^{4}+16a^{4}z^{2}+2a^{4}z^{-2}-8a^{4}+a^{3}z^{9}+10a^{3}z^{7}-19a^{3}z^{5}+4a^{3}z^{3}+6a^{3}z-2a^{3}z^{-1}+4a^{2}z^{8}+a^{2}z^{6}-14a^{2}z^{4}+11a^{2}z^{2}+a^{2}z^{-2}-5a^{2}+6az^{7}-10az^{5}+z^{5}a^{-1}+3az^{3}-z^{3}a^{-1}+4z^{6}-7z^{4}+3z^{2}$ (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

5

1

-1

3

1

4

1

-3

-1

7

3

4

-3

7

6

-1

-5

7

5

2

-7

5

8

3

-9

4

6

-2

-11

1

5

4

-13

1

4

-3

-15

2

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-6$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$