L11a236

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {5u^{2}v^{2}-6u^{2}v+2u^{2}+5uv^{3}-9uv^{2}+5uv+2v^{4}-6v^{3}+5v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {13}{q^{13/2}}}-{\frac {14}{q^{15/2}}}+{\frac {12}{q^{17/2}}}-{\frac {9}{q^{19/2}}}+{\frac {6}{q^{21/2}}}-{\frac {3}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+4a^{11}z+2a^{11}z^{-1}-3a^{9}z^{3}-a^{9}z-5a^{7}z^{3}-5a^{7}z-a^{7}z^{-1}-3a^{5}z^{3}-a^{5}z-a^{3}z^{3}$ (db)
Kauffman polynomial	$a^{14}z^{8}-5a^{14}z^{6}+9a^{14}z^{4}-7a^{14}z^{2}+2a^{14}+3a^{13}z^{9}-15a^{13}z^{7}+24a^{13}z^{5}-13a^{13}z^{3}+2a^{13}z-a^{13}z^{-1}+2a^{12}z^{10}-3a^{12}z^{8}-18a^{12}z^{6}+44a^{12}z^{4}-29a^{12}z^{2}+5a^{12}+10a^{11}z^{9}-42a^{11}z^{7}+51a^{11}z^{5}-21a^{11}z^{3}+7a^{11}z-2a^{11}z^{-1}+2a^{10}z^{10}+7a^{10}z^{8}-50a^{10}z^{6}+64a^{10}z^{4}-25a^{10}z^{2}+3a^{10}+7a^{9}z^{9}-15a^{9}z^{7}-6a^{9}z^{5}+14a^{9}z^{3}-2a^{9}z+11a^{8}z^{8}-28a^{8}z^{6}+15a^{8}z^{4}-2a^{8}z^{2}-a^{8}+12a^{7}z^{7}-27a^{7}z^{5}+17a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}+9a^{6}z^{6}-11a^{6}z^{4}+a^{6}z^{2}+6a^{5}z^{5}-4a^{5}z^{3}+a^{5}z+3a^{4}z^{4}+a^{3}z^{3}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

3

1

-2

-6

3

-8

6

3

-3

-10

7

3

4

-12

6

0

-14

8

7

1

-16

5

7

2

-18

4

7

-3

-20

2

5

3

-22

1

4

-3

-24

2

-26

1

-1

Integral Khovanov Homology

(db, data source)

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