L11a110

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(v^{4}+1\right)}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$q^{25/2}-2q^{23/2}+3q^{21/2}-3q^{19/2}+4q^{17/2}-4q^{15/2}+4q^{13/2}-4q^{11/2}+2q^{9/2}-3q^{7/2}+q^{5/2}-q^{3/2}$ (db)
Signature	7 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-7}+z^{7}a^{-5}-8z^{7}a^{-7}+z^{7}a^{-9}+7z^{5}a^{-5}-23z^{5}a^{-7}+6z^{5}a^{-9}+16z^{3}a^{-5}-31z^{3}a^{-7}+11z^{3}a^{-9}+14za^{-5}-22za^{-7}+8za^{-9}+4a^{-5}z^{-1}-7a^{-7}z^{-1}+3a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{2}a^{-16}+2z^{3}a^{-15}+3z^{4}a^{-14}-3z^{2}a^{-14}+a^{-14}+3z^{5}a^{-13}-4z^{3}a^{-13}+3z^{6}a^{-12}-6z^{4}a^{-12}+3z^{7}a^{-11}-9z^{5}a^{-11}+4z^{3}a^{-11}+3z^{8}a^{-10}-12z^{6}a^{-10}+10z^{4}a^{-10}+3z^{9}a^{-9}-17z^{7}a^{-9}+31z^{5}a^{-9}-25z^{3}a^{-9}+12za^{-9}-3a^{-9}z^{-1}+z^{10}a^{-8}-3z^{8}a^{-8}-7z^{6}a^{-8}+27z^{4}a^{-8}-23z^{2}a^{-8}+7a^{-8}+4z^{9}a^{-7}-28z^{7}a^{-7}+66z^{5}a^{-7}-65z^{3}a^{-7}+30za^{-7}-7a^{-7}z^{-1}+z^{10}a^{-6}-6z^{8}a^{-6}+8z^{6}a^{-6}+8z^{4}a^{-6}-19z^{2}a^{-6}+7a^{-6}+z^{9}a^{-5}-8z^{7}a^{-5}+23z^{5}a^{-5}-30z^{3}a^{-5}+18za^{-5}-4a^{-5}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

26

1

-1

24

1

22

2

1

-1

20

1

0

18

3

2

-1

16

1

0

14

3

0

12

1

0

10

1

3

2

8

2

1

6

1

3

2

4

0

2

1

Integral Khovanov Homology

(db, data source)

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