L11a505

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_10,4,11,3 X_2,18,3,17 X_18,9,19,10 X_20,12,21,11 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13 X_4,19,5,20 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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)\left(u^{2}v^{2}w^{2}-2u^{2}v^{2}w-u^{2}vw^{2}+3u^{2}vw-u^{2}v-u^{2}w+u^{2}-2uv^{2}w^{2}+3uv^{2}w+2uvw^{2}-7uvw+2uv+3uw-2u+v^{2}w^{2}-v^{2}w-vw^{2}+3vw-v-2w+1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+17q^{5}-22q^{4}+27q^{3}-26q^{2}+24q-17+11q^{-1}-5q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}z^{-2}+2z^{6}a^{-4}+6z^{4}a^{-4}+4z^{2}a^{-4}-2a^{-4}z^{-2}-2a^{-4}-z^{8}a^{-2}-4z^{6}a^{-2}-5z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}+z^{6}+2z^{4}$ (db)
Kauffman polynomial	$3z^{10}a^{-2}+3z^{10}a^{-4}+9z^{9}a^{-1}+18z^{9}a^{-3}+9z^{9}a^{-5}+18z^{8}a^{-2}+19z^{8}a^{-4}+11z^{8}a^{-6}+10z^{8}+5az^{7}-11z^{7}a^{-1}-30z^{7}a^{-3}-6z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-57z^{6}a^{-2}-54z^{6}a^{-4}-16z^{6}a^{-6}+4z^{6}a^{-8}-22z^{6}-9az^{5}-8z^{5}a^{-1}+6z^{5}a^{-3}-6z^{5}a^{-5}-10z^{5}a^{-7}+z^{5}a^{-9}-a^{2}z^{4}+46z^{4}a^{-2}+48z^{4}a^{-4}+11z^{4}a^{-6}-5z^{4}a^{-8}+13z^{4}+3az^{3}+8z^{3}a^{-1}+5z^{3}a^{-3}+5z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}-11z^{2}a^{-2}-14z^{2}a^{-4}-3z^{2}a^{-6}+2z^{2}a^{-8}-2z^{2}+2za^{-3}+2za^{-5}-2a^{-2}-3a^{-4}-2a^{-6}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

3

13

6

1

-5

11

3

8

9

12

7

-5

7

15

10

5

13

14

1

3

11

13

-2

1

7

14

7

-1

4

10

-6

-3

1

7

6

-5

4

-4

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$