L10a127

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_18,10,19,9 X_16,12,17,11 X_20,18,11,17 X_4,15,1,16 X_10,20,5,19
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 6, -10}, {7, -2, 4, -5, 9, -7, 8, -6, 10, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)(2vw-v-w+2)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+12q^{5}-15q^{4}+17q^{3}-14q^{2}+13q-7+4q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+z^{2}a^{-2}+z^{2}a^{-4}-z^{2}a^{-6}-z^{2}-a^{-2}+1-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-3}+2z^{9}a^{-5}+5z^{8}a^{-2}+11z^{8}a^{-4}+6z^{8}a^{-6}+6z^{7}a^{-1}+10z^{7}a^{-3}+11z^{7}a^{-5}+7z^{7}a^{-7}-13z^{6}a^{-4}-5z^{6}a^{-6}+4z^{6}a^{-8}+4z^{6}+az^{5}-8z^{5}a^{-1}-20z^{5}a^{-3}-24z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-11z^{4}a^{-2}-z^{4}a^{-4}-3z^{4}a^{-6}-6z^{4}a^{-8}-7z^{4}-az^{3}+2z^{3}a^{-1}+7z^{3}a^{-3}+11z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}+6z^{2}a^{-2}+2z^{2}a^{-4}+2z^{2}a^{-6}+2z^{2}a^{-8}+4z^{2}-za^{-1}-za^{-3}+a^{-2}+a^{-4}+1+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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

χ

17

1

-1

15

3

13

5

1

-4

11

7

3

4

9

8

5

-3

7

9

7

2

5

7

10

3

6

7

-1

1

3

9

6

-1

1

4

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

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