L11a161

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_4,22,5,21 X_14,5,15,6 X_16,13,17,14 X_20,18,21,17 X_18,12,19,11 X_12,20,13,19 X_6,15,1,16
Gauss code	{1, -4, 2, -5, 6, -11}, {4, -1, 3, -2, 9, -10, 7, -6, 11, -7, 8, -9, 10, -8, 5, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+9uv^{3}-13uv^{2}+9uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-8q^{9/2}+{\frac {1}{q^{9/2}}}+14q^{7/2}-{\frac {4}{q^{7/2}}}-19q^{5/2}+{\frac {8}{q^{5/2}}}+21q^{3/2}-{\frac {14}{q^{3/2}}}-q^{13/2}+4q^{11/2}-22{\sqrt {q}}+{\frac {18}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}+2z^{5}a^{-3}-a^{3}z^{3}+5z^{3}a^{-3}-a^{3}z+5za^{-3}+a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+5az^{3}-9z^{3}a^{-1}+5az-9za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}-4az^{9}-8z^{9}a^{-1}-4z^{9}a^{-3}-6a^{2}z^{8}-16z^{8}a^{-2}-7z^{8}a^{-4}-15z^{8}-4a^{3}z^{7}-4az^{7}-4z^{7}a^{-1}-11z^{7}a^{-3}-7z^{7}a^{-5}-a^{4}z^{6}+12a^{2}z^{6}+28z^{6}a^{-2}+3z^{6}a^{-4}-4z^{6}a^{-6}+34z^{6}+10a^{3}z^{5}+28az^{5}+40z^{5}a^{-1}+33z^{5}a^{-3}+10z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-5a^{2}z^{4}-7z^{4}a^{-2}+9z^{4}a^{-4}+6z^{4}a^{-6}-17z^{4}-8a^{3}z^{3}-28az^{3}-40z^{3}a^{-1}-27z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}-a^{4}z^{2}-a^{2}z^{2}-7z^{2}a^{-2}-8z^{2}a^{-4}-3z^{2}a^{-6}-2z^{2}+2a^{3}z+11az+16za^{-1}+9za^{-3}+2za^{-5}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

5

1

4

8

9

3

-6

6

10

5

4

11

9

-2

2

11

10

1

0

8

12

4

-2

6

10

-4

3

9

6

-6

1

5

-4

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$