L11n437

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_13,20,14,21 X_9,18,10,19 X_21,10,22,11 X_19,14,20,7 X_5,17,6,16 X_17,22,18,15 X₂₇₃₈ X_4,11,5,12 X_15,1,16,6
Gauss code	{1, -9, 2, -10, -7, 11}, {9, -1, -4, 5, 10, -2, -3, 6}, {-11, 7, -8, 4, -6, 3, -5, 8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uw+v)\left(uv^{2}w-uv^{2}+u-v^{2}w+w-1\right)}{uv^{3/2}w}}$ (db)
Jones polynomial	$1-q^{-1}+q^{-2}+q^{-4}+q^{-5}+q^{-7}-q^{-8}+q^{-9}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+a^{8}-2a^{6}z^{-2}-3a^{6}-a^{4}z^{6}-5a^{4}z^{4}-5a^{4}z^{2}+a^{4}z^{-2}+a^{2}z^{4}+4a^{2}z^{2}+2a^{2}$ (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 a^{10} z^6-5 a^{10} z^4+5 a^{10} z^2+a^9 z^7-5 a^9 z^5+5 a^9 z^3+a^8 z^8-7 a^8 z^6+16 a^8 z^4-17 a^8 z^2-a^8 z^{-2} +7 a^8+a^7 z^7-7 a^7 z^5+13 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +a^6 z^8-8 a^6 z^6+21 a^6 z^4-25 a^6 z^2-2 a^6 z^{-2} +11 a^6+a^5 z^7-7 a^5 z^5+13 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +a^4 z^6-5 a^4 z^4+3 a^4 z^2-a^4 z^{-2} +3 a^4+a^3 z^7-5 a^3 z^5+5 a^3 z^3+a^2 z^6-5 a^2 z^4+6 a^2 z^2-2 a^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

χ

1

-1

0

-3

1

2

1

0

-5

1

-7

2

4

1

-9

1

2

-11

1

3

1

-13

1

-15

1

0

-17

1

0

-19

1

Integral Khovanov Homology

(db, data source)

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