L11n422

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_3,10,4,11 X_2,18,3,17 X_9,19,10,18 X_20,12,21,11 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13 X_19,5,20,4 X_12,22,7,21
Gauss code	{1, -4, -3, 10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, 4, 5, -10, -6, 11, -8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{2}-2u^{2}v^{2}w-u^{2}vw^{2}+2u^{2}vw-u^{2}v-uv^{2}w^{2}+uv^{2}w+3uvw^{2}-3uvw-uw^{2}+uw+vw^{3}-2vw^{2}+vw+2w^{2}-w}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{9}-2q^{8}+5q^{7}-6q^{6}+8q^{5}-8q^{4}+8q^{3}-5q^{2}+4q-1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}-z^{4}a^{-2}+4z^{4}a^{-4}-2z^{4}a^{-6}-z^{2}a^{-2}+6z^{2}a^{-4}-6z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}+3a^{-4}-6a^{-6}+2a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+5z^{8}a^{-6}+3z^{8}a^{-8}+z^{7}a^{-3}+z^{7}a^{-5}+2z^{7}a^{-7}+2z^{7}a^{-9}-5z^{6}a^{-4}-17z^{6}a^{-6}-11z^{6}a^{-8}+z^{6}a^{-10}+z^{5}a^{-3}-8z^{5}a^{-5}-15z^{5}a^{-7}-6z^{5}a^{-9}+4z^{4}a^{-2}+11z^{4}a^{-4}+25z^{4}a^{-6}+14z^{4}a^{-8}-4z^{4}a^{-10}+z^{3}a^{-1}+14z^{3}a^{-5}+18z^{3}a^{-7}+3z^{3}a^{-9}-3z^{2}a^{-2}-10z^{2}a^{-4}-23z^{2}a^{-6}-12z^{2}a^{-8}+4z^{2}a^{-10}-9za^{-5}-9za^{-7}-a^{-2}+5a^{-4}+11a^{-6}+5a^{-8}-a^{-10}+2a^{-5}z^{-1}+2a^{-7}z^{-1}-a^{-4}z^{-2}-2a^{-6}z^{-2}-a^{-8}z^{-2}$ (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		\

-1

0

1

2

3

4

5

6

7

8

χ

19

1

17

2

1

-1

15

3

13

3

2

-1

11

5

3

2

9

3

0

7

5

0

5

2

5

3

2

3

-1

1

3

-1

1

-1

Integral Khovanov Homology

(db, data source)

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