L11n289

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_11,18,12,19 X_9,21,10,20 X_19,5,20,10 X_15,1,16,4 X_17,22,18,11 X_21,16,22,17
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, 4, -5, -9, 11, -10, 6, -8, 7, -11, 10}

A Braid Representative

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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} , 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 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{u v w^4-2 u v w^3+2 u v w^2-2 u v w-u w^4+u w^3-u w^2+u w-v^2 w^3+v^2 w^2-v^2 w+v^2+2 v w^3-2 v w^2+2 v w-v}{\sqrt{u} v w^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 2 q-2+6 q^{-1} -6 q^{-2} +8 q^{-3} -7 q^{-4} +6 q^{-5} -4 q^{-6} +2 q^{-7} - q^{-8} } (db)
Signature	-2 (db)
HOMFLY-PT 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 -z^4 a^6-3 z^2 a^6-a^6 z^{-2} -3 a^6+z^6 a^4+5 z^4 a^4+11 z^2 a^4+4 a^4 z^{-2} +11 a^4-3 z^4 a^2-11 z^2 a^2-5 a^2 z^{-2} -13 a^2+2 z^2+2 z^{-2} +5} (db)
Kauffman 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 z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-5 z^4 a^8+z^2 a^8+3 z^7 a^7-9 z^5 a^7+8 z^3 a^7-4 z a^7+a^7 z^{-1} +3 z^8 a^6-11 z^6 a^6+17 z^4 a^6-12 z^2 a^6-a^6 z^{-2} +5 a^6+z^9 a^5+z^7 a^5-14 z^5 a^5+31 z^3 a^5-21 z a^5+5 a^5 z^{-1} +5 z^8 a^4-23 z^6 a^4+47 z^4 a^4-40 z^2 a^4-4 a^4 z^{-2} +18 a^4+z^9 a^3-z^7 a^3-7 z^5 a^3+28 z^3 a^3-29 z a^3+9 a^3 z^{-1} +2 z^8 a^2-10 z^6 a^2+28 z^4 a^2-37 z^2 a^2-5 a^2 z^{-2} +21 a^2+z^7 a-3 z^5 a+8 z^3 a-13 z a+5 a z^{-1} +3 z^4-10 z^2-2 z^{-2} +9} (db)

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

2

1

0

-1

5

1

4

-3

3

0

-5

5

3

2

-7

3

4

1

-9

3

4

-1

-11

1

3

2

-13

1

3

-2

-15

1

-17

1

-1

Integral Khovanov Homology

(db, data source)

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