L11a111

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(2v^{4}-v^{3}+2v^{2}-v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{23/2}-2q^{21/2}+4q^{19/2}-6q^{17/2}+9q^{15/2}-10q^{13/2}+9q^{11/2}-9q^{9/2}+6q^{7/2}-5q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+4z^{3}a^{-9}+4za^{-9}+2a^{-9}z^{-1}-z^{7}a^{-7}-5z^{5}a^{-7}-9z^{3}a^{-7}-9za^{-7}-4a^{-7}z^{-1}-z^{7}a^{-5}-4z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+4za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-5z^{9}a^{-7}-3z^{9}a^{-9}-2z^{8}a^{-4}-z^{8}a^{-8}-3z^{8}a^{-10}-z^{7}a^{-3}+6z^{7}a^{-5}+21z^{7}a^{-7}+11z^{7}a^{-9}-3z^{7}a^{-11}+8z^{6}a^{-4}+11z^{6}a^{-6}+12z^{6}a^{-8}+6z^{6}a^{-10}-3z^{6}a^{-12}+5z^{5}a^{-3}-z^{5}a^{-5}-35z^{5}a^{-7}-22z^{5}a^{-9}+5z^{5}a^{-11}-2z^{5}a^{-13}-7z^{4}a^{-4}-18z^{4}a^{-6}-22z^{4}a^{-8}-4z^{4}a^{-10}+6z^{4}a^{-12}-z^{4}a^{-14}-8z^{3}a^{-3}-2z^{3}a^{-5}+35z^{3}a^{-7}+22z^{3}a^{-9}-4z^{3}a^{-11}+3z^{3}a^{-13}-z^{2}a^{-4}+15z^{2}a^{-6}+20z^{2}a^{-8}-4z^{2}a^{-10}-6z^{2}a^{-12}+2z^{2}a^{-14}+5za^{-3}-3za^{-5}-17za^{-7}-10za^{-9}-za^{-11}+a^{-4}-5a^{-6}-6a^{-8}+a^{-10}+2a^{-12}-a^{-3}z^{-1}+a^{-5}z^{-1}+4a^{-7}z^{-1}+2a^{-9}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

1

20

3

1

-2

18

3

1

2

16

6

3

-3

14

4

3

1

12

5

6

1

10

4

0

8

2

5

3

6

3

4

-1

4

1

4

3

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=4$	$i=6$
$r=-2$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=9$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	${\mathbb {Z} }$