L9a41

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a41 at Knotilus!
	[edit L9a41 Quick Notes] L9a41 is $9_{23}^{2}$ in the Rolfsen table of links.

[edit L9a41 Further Notes and Views]

Link Presentations

[edit Notes on L9a41's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_18,5,9,6 X_14,8,15,7 X_16,14,17,13 X_8,16,1,15 X_6,9,7,10 X_4,17,5,18
Gauss code	{1, -2, 3, -9, 4, -8, 5, -7}, {8, -1, 2, -3, 6, -5, 7, -6, 9, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)t(2)+1)(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+4a^{5}z^{3}+5a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-6a^{3}z^{5}-13a^{3}z^{3}-11a^{3}z-a^{3}z^{-1}+az^{5}+4az^{3}+4az$ (db)
Kauffman polynomial	$-z^{3}a^{9}+za^{9}-2z^{4}a^{8}+z^{2}a^{8}-3z^{5}a^{7}+2z^{3}a^{7}-za^{7}-4z^{6}a^{6}+7z^{4}a^{6}-6z^{2}a^{6}-3z^{7}a^{5}+5z^{5}a^{5}-4z^{3}a^{5}+4za^{5}-a^{5}z^{-1}-z^{8}a^{4}-3z^{6}a^{4}+13z^{4}a^{4}-10z^{2}a^{4}+a^{4}-5z^{7}a^{3}+16z^{5}a^{3}-17z^{3}a^{3}+11za^{3}-a^{3}z^{-1}-z^{8}a^{2}+8z^{4}a^{2}-7z^{2}a^{2}-2z^{7}a+8z^{5}a-10z^{3}a+5za-z^{6}+4z^{4}-4z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

1

0

2

1

-1

-2

4

1

3

-4

2

3

1

-6

4

3

1

-8

2

0

-10

2

4

-2

-12

1

3

2

-14

1

-1

-16

1

Integral Khovanov Homology

(db, data source)

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