L11n373

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Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_7,19,8,18 X_21,15,22,14 X_9,20,10,21 X_13,9,14,8 X_15,17,16,22 X_17,5,18,16 X_19,12,20,13 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {-8, 3, -9, 5, -4, 7}, {10, -1, -3, 6, -5, 2, -11, 9, -6, 4, -7, 8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(uv^{2}w^{3}-uv^{2}w^{2}-uvw^{3}+2uvw^{2}-2uvw-uw^{2}+uw+v^{2}w^{2}-v^{2}w-2vw^{2}+2vw-v-w+1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}+3q^{-4}-6q^{3}-5q^{-3}+10q^{2}+9q^{-2}-11q-11q^{-1}+13$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-a^{2}z^{6}-z^{6}a^{-2}-5a^{2}z^{4}-4z^{4}a^{-2}-10a^{2}z^{2}-5z^{2}a^{-2}-2a^{2}z^{-2}-8a^{2}-a^{-2}+z^{8}+6z^{6}+13z^{4}+12z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+5a^{2}z^{8}+4z^{8}a^{-2}+9z^{8}+3a^{3}z^{7}+az^{7}+2z^{7}a^{-1}+4z^{7}a^{-3}-18a^{2}z^{6}-5z^{6}a^{-2}+3z^{6}a^{-4}-26z^{6}-6a^{3}z^{5}-10az^{5}-10z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}+6a^{4}z^{4}+37a^{2}z^{4}-6z^{4}a^{-4}+37z^{4}+7a^{3}z^{3}+18az^{3}+11z^{3}a^{-1}-2z^{3}a^{-3}-2z^{3}a^{-5}-14a^{4}z^{2}-36a^{2}z^{2}+z^{2}a^{-2}+3z^{2}a^{-4}-24z^{2}-7a^{3}z-10az-3za^{-1}+za^{-3}+za^{-5}+7a^{4}+14a^{2}-a^{-2}-a^{-4}+8+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

4

1

-3

5

6

2

4

3

6

5

-1

1

7

5

2

-1

5

7

2

-3

4

6

-2

-5

2

6

4

-7

1

3

-2

-9

3

Integral Khovanov Homology

(db, data source)

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