L11a421

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,7,17,8 X_20,9,21,10 X_8,15,9,16 X_10,19,5,20 X_18,13,19,14 X_22,17,11,18 X_14,21,15,22 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 4, -6}, {11, -2, 7, -9, 5, -3, 8, -7, 6, -4, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {3uv^{2}w^{2}-3uv^{2}w+uv^{2}+3uvw^{3}-7uvw^{2}+5uvw-uv+uw^{4}-4uw^{3}+5uw^{2}-2uw+2v^{2}w^{3}-5v^{2}w^{2}+4v^{2}w-v^{2}+vw^{4}-5vw^{3}+7vw^{2}-3vw-w^{4}+3w^{3}-3w^{2}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-2}-4q^{-3}+10q^{-4}-15q^{-5}+21q^{-6}-22q^{-7}+23q^{-8}-18q^{-9}+14q^{-10}-8q^{-11}+3q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+5a^{12}-9z^{2}a^{10}-5a^{10}z^{-2}-13a^{10}+5z^{4}a^{8}+9z^{2}a^{8}+2a^{8}z^{-2}+7a^{8}+4z^{4}a^{6}+5z^{2}a^{6}+a^{6}+z^{4}a^{4}$ (db)
Kauffman polynomial	$z^{7}a^{15}-4z^{5}a^{15}+6z^{3}a^{15}-4za^{15}+a^{15}z^{-1}+3z^{8}a^{14}-10z^{6}a^{14}+12z^{4}a^{14}-7z^{2}a^{14}-a^{14}z^{-2}+3a^{14}+4z^{9}a^{13}-7z^{7}a^{13}-9z^{5}a^{13}+26z^{3}a^{13}-19za^{13}+5a^{13}z^{-1}+2z^{10}a^{12}+9z^{8}a^{12}-44z^{6}a^{12}+52z^{4}a^{12}-29z^{2}a^{12}-4a^{12}z^{-2}+15a^{12}+13z^{9}a^{11}-23z^{7}a^{11}-17z^{5}a^{11}+48z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+2z^{10}a^{10}+22z^{8}a^{10}-71z^{6}a^{10}+69z^{4}a^{10}-42z^{2}a^{10}-5a^{10}z^{-2}+20a^{10}+9z^{9}a^{9}-35z^{5}a^{9}+35z^{3}a^{9}-18za^{9}+5a^{9}z^{-1}+16z^{8}a^{8}-27z^{6}a^{8}+19z^{4}a^{8}-15z^{2}a^{8}-2a^{8}z^{-2}+8a^{8}+15z^{7}a^{7}-19z^{5}a^{7}+7z^{3}a^{7}+10z^{6}a^{6}-9z^{4}a^{6}+5z^{2}a^{6}-a^{6}+4z^{5}a^{5}+z^{4}a^{4}$ (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

-3

1

-5

4

1

-3

-7

6

-9

9

4

-5

-11

12

6

-13

11

10

-1

-15

12

11

1

-17

8

13

5

-19

6

10

-4

-21

2

8

6

-23

1

6

-5

-25

2

-27

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-8$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-7$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-6$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=-5$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$