L11n298

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_13,19,14,18 X_17,11,18,22 X_7,17,8,16 X_21,8,22,9 X_9,14,10,15 X_15,20,16,21 X_19,5,20,10 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -5, 6, -7, 9}, {-11, 2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{2}w^{3}+3uv^{2}w^{2}-3uv^{2}w+uv^{2}+2uvw^{3}-5uvw^{2}+3uvw-uv-uw^{3}+uw^{2}-v^{2}w^{2}+v^{2}w+vw^{4}-3vw^{3}+5vw^{2}-2vw-w^{4}+3w^{3}-3w^{2}+w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-11q^{3}-2q^{-3}+14q^{2}+7q^{-2}-14q-9q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+z^{2}a^{-4}+a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}-2a^{2}z^{2}-5z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{-2}+3z^{4}+5z^{2}-2z^{-2}+1$ (db)
Kauffman polynomial	$3z^{9}a^{-1}+3z^{9}a^{-3}+14z^{8}a^{-2}+6z^{8}a^{-4}+8z^{8}+6az^{7}+5z^{7}a^{-1}+3z^{7}a^{-3}+4z^{7}a^{-5}+a^{2}z^{6}-40z^{6}a^{-2}-15z^{6}a^{-4}+z^{6}a^{-6}-23z^{6}-11az^{5}-28z^{5}a^{-1}-27z^{5}a^{-3}-10z^{5}a^{-5}+8a^{2}z^{4}+39z^{4}a^{-2}+8z^{4}a^{-4}-2z^{4}a^{-6}+37z^{4}+3a^{3}z^{3}+14az^{3}+30z^{3}a^{-1}+25z^{3}a^{-3}+6z^{3}a^{-5}-9a^{2}z^{2}-20z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-25z^{2}-a^{3}z-3az-7za^{-1}-7za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

5

1

4

7

6

3

-3

5

8

5

3

7

0

1

7

0

-1

4

9

5

-3

3

5

-2

-5

5

-7

2

-2

Integral Khovanov Homology

(db, data source)

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