L11n431

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,10,21,9 X_5,15,6,14 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_10,4,11,3 X_18,11,19,12 X_15,1,16,6 X_4,20,5,19 X_2,21,3,22
Gauss code	{1, -11, 7, -10, -3, 9}, {5, -1, 2, -7, 8, -4}, {4, 3, -9, -5, 6, -8, 10, -2, 11, -6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)\left(-u^{2}v^{2}w+u^{2}v^{2}+u^{2}vw-u^{2}v+uv^{2}w-uv^{2}+uvw^{2}-3uvw+uv-uw^{2}+uw-vw^{2}+vw+w^{2}-w\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+7q^{7}-9q^{6}+12q^{5}-11q^{4}+11q^{3}-7q^{2}+5q-1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}-2z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}z^{-2}+z^{6}a^{-4}+3z^{4}a^{-4}+3z^{2}a^{-4}-2a^{-4}z^{-2}-2a^{-4}-z^{4}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-2z^{4}a^{-10}+4z^{7}a^{-9}-11z^{5}a^{-9}+5z^{3}a^{-9}+6z^{8}a^{-8}-19z^{6}a^{-8}+15z^{4}a^{-8}-4z^{2}a^{-8}+3z^{9}a^{-7}-3z^{7}a^{-7}-10z^{5}a^{-7}+8z^{3}a^{-7}+11z^{8}a^{-6}-36z^{6}a^{-6}+37z^{4}a^{-6}-11z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+3z^{9}a^{-5}-5z^{7}a^{-5}+z^{5}a^{-5}+2z^{3}a^{-5}+2za^{-5}-2a^{-5}z^{-1}+5z^{8}a^{-4}-16z^{6}a^{-4}+25z^{4}a^{-4}-10z^{2}a^{-4}+2a^{-4}z^{-2}-3a^{-4}+2z^{7}a^{-3}+2za^{-3}-2a^{-3}z^{-1}+5z^{4}a^{-2}-3z^{2}a^{-2}+a^{-2}z^{-2}-2a^{-2}+z^{3}a^{-1}$ (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		\

-1

0

1

2

3

4

5

6

7

8

χ

19

1

17

3

-3

15

4

1

3

13

5

3

-2

11

7

4

3

9

5

6

1

7

6

0

5

3

7

4

3

2

4

-2

1

4

-1

1

-1

Integral Khovanov Homology

(db, data source)

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