L11a48

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_20,16,21,15 X_16,20,17,19 X_4,21,1,22 X_12,6,13,5 X_10,4,11,3 X_22,12,5,11 X_14,9,15,10 X_2,14,3,13 X_8,17,9,18
Gauss code	{1, -10, 7, -5}, {6, -1, 2, -11, 9, -7, 8, -6, 10, -9, 3, -4, 11, -2, 4, -3, 5, -8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-4v+1\right)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-10q^{9/2}+15q^{7/2}-20q^{5/2}+24q^{3/2}-23{\sqrt {q}}+{\frac {19}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+4az^{3}-4z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+az-2za^{-1}+3za^{-3}-za^{-5}+a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-4z^{4}a^{-6}+9z^{7}a^{-5}-15z^{5}a^{-5}+10z^{3}a^{-5}-5za^{-5}+a^{-5}z^{-1}+11z^{8}a^{-4}+a^{4}z^{6}-19z^{6}a^{-4}-2a^{4}z^{4}+14z^{4}a^{-4}+a^{4}z^{2}-4z^{2}a^{-4}+a^{-4}+7z^{9}a^{-3}+4a^{3}z^{7}-10a^{3}z^{5}-19z^{5}a^{-3}+8a^{3}z^{3}+23z^{3}a^{-3}-a^{3}z-13za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+2z^{10}a^{-2}+6a^{2}z^{8}+16z^{8}a^{-2}-11a^{2}z^{6}-38z^{6}a^{-2}+3a^{2}z^{4}+26z^{4}a^{-2}+a^{2}z^{2}-6z^{2}a^{-2}+a^{2}+3a^{-2}+5az^{9}+12z^{9}a^{-1}-az^{7}-14z^{7}a^{-1}-16az^{5}-9z^{5}a^{-1}+14az^{3}+18z^{3}a^{-1}-3az-10za^{-1}-az^{-1}+a^{-1}z^{-1}+2z^{10}+11z^{8}-27z^{6}+13z^{4}-2z^{2}+2$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

7

1

6

8

3

-5

6

12

7

5

4

12

8

-4

2

11

12

-1

0

10

14

4

-2

5

9

-4

3

10

7

-6

1

5

-4

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$