L11a426

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_20,11,21,12 X_22,17,11,18 X_16,21,17,22 X_10,13,5,14 X_8,20,9,19 X_18,8,19,7 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, {4, -2, 7, -3, 11, -6, 5, -9, 8, -4, 6, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{4}-3uv^{2}w^{3}+4uv^{2}w^{2}-2uv^{2}w-uvw^{4}+5uvw^{3}-8uvw^{2}+6uvw-uv-uw^{3}+3uw^{2}-3uw+u-v^{2}w^{4}+3v^{2}w^{3}-3v^{2}w^{2}+v^{2}w+vw^{4}-6vw^{3}+8vw^{2}-5vw+v+2w^{3}-4w^{2}+3w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-8}+4q^{-7}-10q^{-6}+16q^{-5}-22q^{-4}+q^{3}+26q^{-3}-4q^{2}-24q^{-2}+10q+23q^{-1}-15$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+7a^{4}z^{4}-11a^{2}z^{4}+3z^{4}-2a^{6}z^{2}+10a^{4}z^{2}-13a^{2}z^{2}+4z^{2}-2a^{6}+8a^{4}-10a^{2}+4-a^{6}z^{-2}+4a^{4}z^{-2}-5a^{2}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$a^{9}z^{5}-a^{9}z^{3}+4a^{8}z^{6}-4a^{8}z^{4}+a^{8}z^{2}+9a^{7}z^{7}-14a^{7}z^{5}+11a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+11a^{6}z^{8}-15a^{6}z^{6}+9a^{6}z^{4}-5a^{6}z^{2}-a^{6}z^{-2}+3a^{6}+7a^{5}z^{9}+7a^{5}z^{7}-37a^{5}z^{5}+38a^{5}z^{3}-18a^{5}z+5a^{5}z^{-1}+2a^{4}z^{10}+22a^{4}z^{8}-56a^{4}z^{6}+51a^{4}z^{4}-28a^{4}z^{2}-4a^{4}z^{-2}+12a^{4}+14a^{3}z^{9}-13a^{3}z^{7}-23a^{3}z^{5}+36a^{3}z^{3}-24a^{3}z+9a^{3}z^{-1}+2a^{2}z^{10}+19a^{2}z^{8}-57a^{2}z^{6}+z^{6}a^{-2}+58a^{2}z^{4}-2z^{4}a^{-2}-35a^{2}z^{2}+z^{2}a^{-2}-5a^{2}z^{-2}+15a^{2}+7az^{9}-7az^{7}+4z^{7}a^{-1}-9az^{5}-8z^{5}a^{-1}+14az^{3}+4z^{3}a^{-1}-11az+5az^{-1}+8z^{8}-19z^{6}+18z^{4}-12z^{2}-2z^{-2}+7$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

7

1

6

1

9

4

-5

-1

14

6

8

-3

12

11

-1

-5

14

12

2

-7

9

13

4

-9

7

13

-6

-11

3

9

6

-13

1

7

-6

-15

3

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$