L11a507

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,12,19,11 X_10,4,11,3 X_2,20,3,19 X_16,8,17,7 X_20,9,21,10 X_12,18,7,17 X_22,16,13,15 X_14,6,15,5 X_4,14,5,13 X_6,21,1,22
Gauss code	{1, -4, 3, -10, 9, -11}, {5, -1, 6, -3, 2, -7}, {10, -9, 8, -5, 7, -2, 4, -6, 11, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(u^{2}v^{2}w+u^{2}\left(-v^{2}\right)+u^{2}vw^{2}-2u^{2}vw+u^{2}v-u^{2}w^{2}+u^{2}w-2uv^{2}w+2uv^{2}-2uvw^{2}+5uvw-2uv+2uw^{2}-2uw+v^{2}w-v^{2}+vw^{2}-2vw+v-w^{2}+w\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+8q^{7}-13q^{6}+19q^{5}-20q^{4}+22q^{3}-18q^{2}-q^{-2}+14q+4q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}-2z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}z^{-2}+z^{6}a^{-4}+z^{4}a^{-4}-2a^{-4}z^{-2}-2a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$2z^{10}a^{-4}+2z^{10}a^{-6}+6z^{9}a^{-3}+12z^{9}a^{-5}+6z^{9}a^{-7}+8z^{8}a^{-2}+12z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}a^{-8}+7z^{7}a^{-1}-3z^{7}a^{-3}-24z^{7}a^{-5}-10z^{7}a^{-7}+4z^{7}a^{-9}-10z^{6}a^{-2}-35z^{6}a^{-4}-41z^{6}a^{-6}-19z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-10z^{5}a^{-1}-6z^{5}a^{-3}+14z^{5}a^{-5}-z^{5}a^{-7}-10z^{5}a^{-9}+2z^{4}a^{-2}+34z^{4}a^{-4}+44z^{4}a^{-6}+16z^{4}a^{-8}-2z^{4}a^{-10}-6z^{4}-az^{3}+3z^{3}a^{-1}+4z^{3}a^{-3}+5z^{3}a^{-7}+5z^{3}a^{-9}+z^{2}a^{-2}-10z^{2}a^{-4}-15z^{2}a^{-6}-6z^{2}a^{-8}+2z^{2}+2za^{-3}+2za^{-5}-2a^{-2}-3a^{-4}-2a^{-6}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

19

1

17

3

-3

15

5

1

4

13

8

3

-5

11

5

6

9

10

9

-1

7

12

10

2

5

8

12

4

3

6

10

-4

1

3

9

6

-1

1

5

-4

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$