L9a48

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	[edit L9a48 Quick Notes] L9a48 is $9_{5}^{3}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9a48's Link Presentations]

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

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

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} , $v$ , $w$ , ...)	${\frac {uv^{2}w^{2}-uv^{2}w-uvw^{2}+2uvw-uv-uw+2u-2v^{2}w^{2}+v^{2}w+vw^{2}-2vw+v+w-1}{{\sqrt {u}}vw}}$ (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 -q^6+3 q^5-4 q^4+6 q^3-6 q^2+6 q-4+4 q^{-1} - q^{-2} + q^{-3} } (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^{-4} -2 z^2 a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+3 a^{-2} -2 z^4-7 z^2-2 z^{-2} -6} (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^8 a^{-2} +z^8+a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+2 z^6 a^{-2} +4 z^6 a^{-4} -z^6-2 a z^5-10 z^5 a^{-1} -4 z^5 a^{-3} +4 z^5 a^{-5} -5 a^2 z^4-8 z^4 a^{-2} -4 z^4 a^{-4} +3 z^4 a^{-6} -6 z^4-3 a z^3+2 z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +8 a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-6} +11 z^2+6 a z+6 z a^{-1} -5 a^2-3 a^{-2} + a^{-4} -8-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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 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} ).

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

13

1

-1

11

2

9

2

1

-1

7

4

2

5

4

0

3

2

0

1

3

5

2

-1

1

0

-3

3

-5

1

0

-7

1

Integral Khovanov Homology

(db, data source)

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