L11a55

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,8,15,7 X_22,16,5,15 X_16,9,17,10 X_8,21,9,22 X_20,17,21,18 X_18,12,19,11 X_10,20,11,19 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 8, -2, 11, -3, 4, -5, 7, -8, 9, -7, 6, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+7v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-9q^{9/2}+{\frac {1}{q^{9/2}}}+14q^{7/2}-{\frac {4}{q^{7/2}}}-19q^{5/2}+{\frac {8}{q^{5/2}}}+22q^{3/2}-{\frac {14}{q^{3/2}}}-q^{13/2}+4q^{11/2}-22{\sqrt {q}}+{\frac {18}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-9z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+5az-9za^{-1}+6za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}-4az^{9}-9z^{9}a^{-1}-5z^{9}a^{-3}-6a^{2}z^{8}-20z^{8}a^{-2}-9z^{8}a^{-4}-17z^{8}-4a^{3}z^{7}-5az^{7}-z^{7}a^{-1}-8z^{7}a^{-3}-8z^{7}a^{-5}-a^{4}z^{6}+12a^{2}z^{6}+50z^{6}a^{-2}+13z^{6}a^{-4}-4z^{6}a^{-6}+46z^{6}+10a^{3}z^{5}+34az^{5}+45z^{5}a^{-1}+35z^{5}a^{-3}+13z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-5a^{2}z^{4}-45z^{4}a^{-2}-8z^{4}a^{-4}+5z^{4}a^{-6}-39z^{4}-8a^{3}z^{3}-37az^{3}-58z^{3}a^{-1}-38z^{3}a^{-3}-8z^{3}a^{-5}+z^{3}a^{-7}-a^{4}z^{2}+a^{2}z^{2}+17z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}+14z^{2}+3a^{3}z+15az+26za^{-1}+18za^{-3}+4za^{-5}-a^{2}-3a^{-2}-a^{-4}-2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

6

1

5

8

3

-5

6

11

6

5

4

11

8

-3

2

11

0

9

13

4

-2

5

9

-4

3

9

6

-6

1

5

-4

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$