L11a231

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,13,7,14 X_14,9,15,10 X_10,21,11,22 X_18,5,19,6 X_20,16,21,15 X_16,20,17,19 X₂₇₃₈ X_4,11,5,12 X_6,17,1,18
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 4, -5, 10, -2, 3, -4, 7, -8, 11, -6, 8, -7, 5, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{3}-7u^{2}v^{2}+6u^{2}v-2u^{2}+uv^{4}-8uv^{3}+13uv^{2}-8uv+u-2v^{4}+6v^{3}-7v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	${\frac {21}{q^{9/2}}}-{\frac {19}{q^{7/2}}}+{\frac {14}{q^{5/2}}}-{\frac {9}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {13}{q^{15/2}}}+{\frac {17}{q^{13/2}}}-{\frac {21}{q^{11/2}}}-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-5a^{7}z^{3}-6a^{7}z-2a^{7}z^{-1}+2a^{5}z^{5}+2a^{5}z^{3}+a^{5}z+a^{3}z^{5}-a^{3}z^{3}-2a^{3}z-az^{3}$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-8a^{11}z^{5}+7a^{11}z^{3}-3a^{11}z+a^{11}z^{-1}+4a^{10}z^{8}-5a^{10}z^{6}-6a^{10}z^{4}+9a^{10}z^{2}-3a^{10}+4a^{9}z^{9}-2a^{9}z^{7}-11a^{9}z^{5}+15a^{9}z^{3}-12a^{9}z+3a^{9}z^{-1}+2a^{8}z^{10}+6a^{8}z^{8}-17a^{8}z^{6}+8a^{8}z^{4}+3a^{8}z^{2}-3a^{8}+11a^{7}z^{9}-20a^{7}z^{7}+9a^{7}z^{5}+6a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+12a^{6}z^{8}-34a^{6}z^{6}+31a^{6}z^{4}-8a^{6}z^{2}+7a^{5}z^{9}-7a^{5}z^{7}-2a^{5}z^{5}+5a^{5}z^{3}+10a^{4}z^{8}-19a^{4}z^{6}+15a^{4}z^{4}-5a^{4}z^{2}+8a^{3}z^{7}-13a^{3}z^{5}+6a^{3}z^{3}-2a^{3}z+4a^{2}z^{6}-5a^{2}z^{4}+az^{5}-az^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

3

-3

-2

6

1

5

-4

9

4

-5

-6

10

5

-8

11

9

-2

-10

10

0

-12

7

11

4

-14

6

10

-4

-16

2

8

6

-18

1

5

-4

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$