L11n415

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Planar diagram presentation	X₈₁₉₂ X_5,15,6,14 X_10,3,11,4 X_13,5,14,4 X₂₇₃₈ X_6,9,1,10 X_18,12,19,11 X_12,18,7,17 X_20,16,21,15 X_22,20,13,19 X_16,22,17,21
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, 7, -8}, {-4, 2, 9, -11, 8, -7, 10, -9, 11, -10}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}vw^{2}-2u^{2}vw+u^{2}v-u^{2}w^{2}+u^{2}w+2uv^{2}w^{2}-2uv^{2}w+uv^{2}+uvw^{3}-4uvw^{2}+4uvw-uv-uw^{3}+2uw^{2}-2uw-v^{2}w^{2}+v^{2}w-vw^{3}+2vw^{2}-2vw}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+9q^{5}-11q^{4}+12q^{3}-10q^{2}+9q+3q^{-1}-4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-a^{-8}+3z^{2}a^{-6}-a^{-6}z^{-2}+2a^{-6}-2z^{4}a^{-4}-z^{2}a^{-4}+4a^{-4}z^{-2}+4a^{-4}-3z^{4}a^{-2}-8z^{2}a^{-2}-5a^{-2}z^{-2}-10a^{-2}+3z^{2}+2z^{-2}+5$ (db)
Kauffman polynomial	$z^{9}a^{-3}+z^{9}a^{-5}+4z^{8}a^{-2}+7z^{8}a^{-4}+3z^{8}a^{-6}+3z^{7}a^{-1}+7z^{7}a^{-3}+8z^{7}a^{-5}+4z^{7}a^{-7}-14z^{6}a^{-2}-17z^{6}a^{-4}+3z^{6}a^{-8}-9z^{5}a^{-1}-28z^{5}a^{-3}-26z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}+32z^{4}a^{-2}+23z^{4}a^{-4}-9z^{4}a^{-6}-6z^{4}a^{-8}+6z^{4}+18z^{3}a^{-1}+48z^{3}a^{-3}+33z^{3}a^{-5}+z^{3}a^{-7}-2z^{3}a^{-9}-38z^{2}a^{-2}-19z^{2}a^{-4}+6z^{2}a^{-6}+3z^{2}a^{-8}-16z^{2}-19za^{-1}-35za^{-3}-19za^{-5}-2za^{-7}+za^{-9}+22a^{-2}+13a^{-4}-a^{-8}+11+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

2

13

4

1

-3

11

5

2

3

9

7

5

-2

7

5

4

1

5

7

2

3

4

5

-1

1

6

5

-1

2

3

-1

-3

3

Integral Khovanov Homology

(db, data source)

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