L11n268

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(3)^{5}+t(1)t(3)^{4}-2t(1)t(2)t(3)^{4}+t(2)t(3)^{4}-2t(3)^{4}-t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+2t(3)^{3}+t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-2t(3)^{2}-t(1)t(3)+2t(1)t(2)t(3)-t(2)t(3)+2t(3)-t(1)t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{11}+2q^{10}-5q^{9}+6q^{8}-8q^{7}+10q^{6}-7q^{5}+7q^{4}-3q^{3}+3q^{2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-10}-2a^{-10}z^{-2}-3a^{-10}+3z^{4}a^{-8}+12z^{2}a^{-8}+7a^{-8}z^{-2}+15a^{-8}-2z^{6}a^{-6}-11z^{4}a^{-6}-23z^{2}a^{-6}-8a^{-6}z^{-2}-22a^{-6}+3z^{4}a^{-4}+11z^{2}a^{-4}+3a^{-4}z^{-2}+10a^{-4}$ (db)
Kauffman polynomial	$z^{5}a^{-13}-3z^{3}a^{-13}+3za^{-13}-a^{-13}z^{-1}+2z^{6}a^{-12}-4z^{4}a^{-12}+a^{-12}+2z^{7}a^{-11}-z^{5}a^{-11}-6z^{3}a^{-11}+3za^{-11}-a^{-11}z^{-1}+2z^{8}a^{-10}-3z^{6}a^{-10}+4z^{4}a^{-10}-9z^{2}a^{-10}-2a^{-10}z^{-2}+7a^{-10}+z^{9}a^{-9}+2z^{7}a^{-9}-13z^{5}a^{-9}+26z^{3}a^{-9}-21za^{-9}+7a^{-9}z^{-1}+6z^{8}a^{-8}-24z^{6}a^{-8}+44z^{4}a^{-8}-36z^{2}a^{-8}-7a^{-8}z^{-2}+22a^{-8}+z^{9}a^{-7}+3z^{7}a^{-7}-23z^{5}a^{-7}+52z^{3}a^{-7}-45za^{-7}+15a^{-7}z^{-1}+4z^{8}a^{-6}-19z^{6}a^{-6}+42z^{4}a^{-6}-46z^{2}a^{-6}-8a^{-6}z^{-2}+28a^{-6}+3z^{7}a^{-5}-12z^{5}a^{-5}+23z^{3}a^{-5}-24za^{-5}+8a^{-5}z^{-1}+6z^{4}a^{-4}-19z^{2}a^{-4}-3a^{-4}z^{-2}+13a^{-4}$ (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		\

0

1

2

3

4

5

6

7

8

9

χ

23

1

-1

21

1

19

4

1

-3

17

2

1

15

6

4

-2

13

4

2

11

4

7

3

9

3

0

7

4

5

3

0

3

Integral Khovanov Homology

(db, data source)

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