L11n283

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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_9,18,10,19 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, -6, 11}, {-8, -2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+6q^{4}-9q^{3}-q^{-3}+11q^{2}+5q^{-2}-10q-6q^{-1}+11$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+z^{2}a^{-4}-a^{-4}z^{-2}-a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}-a^{2}z^{2}-2z^{2}a^{-2}+2a^{2}z^{-2}+4a^{-2}z^{-2}+a^{2}+4a^{-2}+2z^{4}+2z^{2}-5z^{-2}-4$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+2az^{7}+2z^{7}a^{-3}+4z^{7}a^{-5}-28z^{6}a^{-2}-15z^{6}a^{-4}+z^{6}a^{-6}-12z^{6}-az^{5}-8z^{5}a^{-1}-19z^{5}a^{-3}-12z^{5}a^{-5}+5a^{2}z^{4}+29z^{4}a^{-2}+11z^{4}a^{-4}-2z^{4}a^{-6}+21z^{4}+a^{3}z^{3}+az^{3}+6z^{3}a^{-1}+13z^{3}a^{-3}+7z^{3}a^{-5}-3a^{2}z^{2}-12z^{2}a^{-2}-4z^{2}a^{-4}-11z^{2}+5az+9za^{-1}+5za^{-3}+za^{-5}-3a^{2}-2a^{-2}-4-5az^{-1}-9a^{-1}z^{-1}-5a^{-3}z^{-1}-a^{-5}z^{-1}+2a^{2}z^{-2}+4a^{-2}z^{-2}+a^{-4}z^{-2}+5z^{-2}$ (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

χ

13

1

11

3

-3

9

3

1

2

7

6

3

-3

5

3

2

3

5

6

1

6

5

1

-1

2

7

5

-3

3

4

-1

-5

4

-7

1

-1

Integral Khovanov Homology

(db, data source)

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