L11n289

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_11,18,12,19 X_9,21,10,20 X_19,5,20,10 X_15,1,16,4 X_17,22,18,11 X_21,16,22,17
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, 4, -5, -9, 11, -10, 6, -8, 7, -11, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{4}-2uvw^{3}+2uvw^{2}-2uvw-uw^{4}+uw^{3}-uw^{2}+uw-v^{2}w^{3}+v^{2}w^{2}-v^{2}w+v^{2}+2vw^{3}-2vw^{2}+2vw-v}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$2q-2+6q^{-1}-6q^{-2}+8q^{-3}-7q^{-4}+6q^{-5}-4q^{-6}+2q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-a^{6}z^{-2}-3a^{6}+z^{6}a^{4}+5z^{4}a^{4}+11z^{2}a^{4}+4a^{4}z^{-2}+11a^{4}-3z^{4}a^{2}-11z^{2}a^{2}-5a^{2}z^{-2}-13a^{2}+2z^{2}+2z^{-2}+5$ (db)
Kauffman polynomial	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-5z^{4}a^{8}+z^{2}a^{8}+3z^{7}a^{7}-9z^{5}a^{7}+8z^{3}a^{7}-4za^{7}+a^{7}z^{-1}+3z^{8}a^{6}-11z^{6}a^{6}+17z^{4}a^{6}-12z^{2}a^{6}-a^{6}z^{-2}+5a^{6}+z^{9}a^{5}+z^{7}a^{5}-14z^{5}a^{5}+31z^{3}a^{5}-21za^{5}+5a^{5}z^{-1}+5z^{8}a^{4}-23z^{6}a^{4}+47z^{4}a^{4}-40z^{2}a^{4}-4a^{4}z^{-2}+18a^{4}+z^{9}a^{3}-z^{7}a^{3}-7z^{5}a^{3}+28z^{3}a^{3}-29za^{3}+9a^{3}z^{-1}+2z^{8}a^{2}-10z^{6}a^{2}+28z^{4}a^{2}-37z^{2}a^{2}-5a^{2}z^{-2}+21a^{2}+z^{7}a-3z^{5}a+8z^{3}a-13za+5az^{-1}+3z^{4}-10z^{2}-2z^{-2}+9$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

2

1

0

-1

5

1

4

-3

3

0

-5

5

3

2

-7

3

4

1

-9

3

4

-1

-11

1

3

2

-13

1

3

-2

-15

1

-17

1

-1

Integral Khovanov Homology

(db, data source)

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