L11a127

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

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Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , $v$ , $w$ , ...)	$-{\frac {3uv^{4}-10uv^{3}+14uv^{2}-7uv+u+v^{5}-7v^{4}+14v^{3}-10v^{2}+3v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {19}{q^{9/2}}}+{\frac {22}{q^{7/2}}}+q^{5/2}-{\frac {23}{q^{5/2}}}-4q^{3/2}+{\frac {20}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {14}{q^{11/2}}}+9{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-4za^{7}-3a^{7}z^{-1}+6z^{3}a^{5}+9za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-7z^{3}a^{3}-8za^{3}-2a^{3}z^{-1}-z^{5}a+z^{3}a+za+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+6a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-10a^{8}z^{6}+12a^{8}z^{4}-6a^{8}z^{2}+a^{8}+4a^{7}z^{9}-7a^{7}z^{7}-8a^{7}z^{5}+23a^{7}z^{3}-15a^{7}z+3a^{7}z^{-1}+2a^{6}z^{10}+9a^{6}z^{8}-42a^{6}z^{6}+44a^{6}z^{4}-17a^{6}z^{2}+3a^{6}+13a^{5}z^{9}-22a^{5}z^{7}-19a^{5}z^{5}+44a^{5}z^{3}-24a^{5}z+4a^{5}z^{-1}+2a^{4}z^{10}+22a^{4}z^{8}-68a^{4}z^{6}+53a^{4}z^{4}-15a^{4}z^{2}+2a^{4}+9a^{3}z^{9}+a^{3}z^{7}-41a^{3}z^{5}+42a^{3}z^{3}-16a^{3}z+2a^{3}z^{-1}+16a^{2}z^{8}-27a^{2}z^{6}+14a^{2}z^{4}+z^{4}a^{-2}-3a^{2}z^{2}+a^{2}+15az^{7}-22az^{5}+4z^{5}a^{-1}+14az^{3}-z^{3}a^{-1}-3az+9z^{6}-6z^{4}+z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

6

1

-5

0

10

3

7

-2

11

7

-4

12

9

3

-6

10

11

1

-8

9

12

-3

-10

6

11

5

-12

2

8

-6

-14

1

6

5

-16

2

-2

-18

1

Integral Khovanov Homology

(db, data source)

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