L11n316

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,17,8,16 X_20,10,21,9 X_18,12,19,11 X_22,20,11,19 X_15,9,16,8 X_10,22,5,21 X_17,14,18,15 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 $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{2}w-uv^{2}+uvw^{2}-3uvw+2uv-uw^{2}+uw-v^{2}w+v^{2}-2vw^{2}+3vw-v+w^{2}-2w}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{9}-3q^{8}+5q^{7}-6q^{6}+8q^{5}-7q^{4}+7q^{3}-4q^{2}+3q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}-z^{4}a^{-6}+a^{-6}z^{-2}+2a^{-6}-2z^{4}a^{-4}-5z^{2}a^{-4}-2a^{-4}z^{-2}-6a^{-4}+3z^{2}a^{-2}+a^{-2}z^{-2}+4a^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}-2z^{7}a^{-5}+z^{7}a^{-7}+3z^{7}a^{-9}-z^{6}a^{-4}-11z^{6}a^{-6}-9z^{6}a^{-8}+z^{6}a^{-10}+3z^{5}a^{-3}+5z^{5}a^{-5}-8z^{5}a^{-7}-10z^{5}a^{-9}-z^{4}a^{-4}+8z^{4}a^{-6}+6z^{4}a^{-8}-3z^{4}a^{-10}-4z^{3}a^{-3}-8z^{3}a^{-5}+3z^{3}a^{-7}+7z^{3}a^{-9}+5z^{2}a^{-2}+10z^{2}a^{-4}-3z^{2}a^{-8}+2z^{2}a^{-10}+6za^{-3}+6za^{-5}-5a^{-2}-8a^{-4}-3a^{-6}+a^{-8}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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		\

0

1

2

3

4

5

6

7

8

χ

19

1

17

2

-2

15

3

1

2

13

4

3

-1

11

4

2

9

3

4

1

7

4

0

5

1

4

3

2

3

-1

1

3

Integral Khovanov Homology

(db, data source)

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