L11a43

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

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Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

3

1

-2

6

2

4

5

3

-2

2

8

6

2

0

6

7

1

-2

4

6

-2

-4

4

6

2

-6

1

4

-3

-8

1

4

3

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	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 i=2}
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 r=-6}	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}}
$r=-5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	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}^{6}\oplus{\mathbb Z}_2^{4}}	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}^{4}}
$r=-1$	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}^{6}\oplus{\mathbb Z}_2^{6}}	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=2$	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}^{3}\oplus{\mathbb Z}_2^{6}}	${\mathbb {Z} }^{6}$
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 r=3}	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
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 r=4}	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}\oplus{\mathbb Z}_2^{2}}	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	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}}