L11n119

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,15,4,14 X_16,10,17,9 X_11,21,12,20 X_21,9,22,8 X_7,19,8,18 X_19,13,20,12 X_10,16,11,15 X_17,5,18,22 X₂₅₃₆ X_13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -6, 5, 3, -8, -4, 7, -11, 2, 8, -3, -9, 6, -7, 4, -5, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(2v^{2}-v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$7q^{9/2}-7q^{7/2}+4q^{5/2}-3q^{3/2}+2q^{17/2}-4q^{15/2}+6q^{13/2}-7q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-5}+3z^{3}a^{-3}-8z^{3}a^{-5}+2z^{3}a^{-7}+7za^{-3}-11za^{-5}+4za^{-7}+3a^{-3}z^{-1}-5a^{-5}z^{-1}+2a^{-7}z^{-1}$ (db)
Kauffman polynomial	$3z^{4}a^{-10}-5z^{2}a^{-10}+a^{-10}+z^{7}a^{-9}+z^{5}a^{-9}-4z^{3}a^{-9}+z^{8}a^{-8}+z^{6}a^{-8}-3z^{4}a^{-8}+5z^{7}a^{-7}-11z^{5}a^{-7}+12z^{3}a^{-7}-7za^{-7}+2a^{-7}z^{-1}+z^{8}a^{-6}+4z^{6}a^{-6}-12z^{4}a^{-6}+14z^{2}a^{-6}-5a^{-6}+4z^{7}a^{-5}-12z^{5}a^{-5}+22z^{3}a^{-5}-17za^{-5}+5a^{-5}z^{-1}+3z^{6}a^{-4}-6z^{4}a^{-4}+9z^{2}a^{-4}-5a^{-4}+6z^{3}a^{-3}-10za^{-3}+3a^{-3}z^{-1}$ (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		\

0

1

2

3

4

5

6

7

χ

18

2

-2

16

2

14

4

2

-2

12

3

2

1

10

4

0

8

3

0

6

1

4

3

4

2

3

-1

2

3

Integral Khovanov Homology

(db, data source)

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