L11a391

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,7,17,8 X_8,15,5,16 X_18,11,19,12 X_22,17,9,18 X_20,13,21,14 X_12,19,13,20 X_14,21,15,22 X₂₅₃₆ X_4,9,1,10
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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{5}+t(2)t(3)^{5}-t(3)^{5}-3t(1)t(3)^{4}+2t(1)t(2)t(3)^{4}-3t(2)t(3)^{4}+2t(3)^{4}+3t(1)t(3)^{3}-2t(1)t(2)t(3)^{3}+3t(2)t(3)^{3}-2t(3)^{3}-3t(1)t(3)^{2}+2t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+2t(3)^{2}+3t(1)t(3)-2t(1)t(2)t(3)+3t(2)t(3)-2t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{-14}+2q^{-13}-6q^{-12}+9q^{-11}-12q^{-10}+15q^{-9}-14q^{-8}+14q^{-7}-8q^{-6}+7q^{-5}-3q^{-4}+q^{-3}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-2a^{14}z^{-2}-a^{14}+4a^{12}z^{2}+7a^{12}z^{-2}+12a^{12}-6a^{10}z^{4}-23a^{10}z^{2}-8a^{10}z^{-2}-24a^{10}+3a^{8}z^{6}+14a^{8}z^{4}+21a^{8}z^{2}+3a^{8}z^{-2}+13a^{8}+a^{6}z^{6}+3a^{6}z^{4}+a^{6}z^{2}$ (db)
Kauffman polynomial	$a^{17}z^{5}-3a^{17}z^{3}+3a^{17}z-a^{17}z^{-1}+2a^{16}z^{6}-3a^{16}z^{4}+a^{16}+3a^{15}z^{7}-3a^{15}z^{5}-2a^{15}z^{3}+3a^{15}z-a^{15}z^{-1}+3a^{14}z^{8}-2a^{14}z^{6}+2a^{14}z^{4}-9a^{14}z^{2}-2a^{14}z^{-2}+7a^{14}+2a^{13}z^{9}+3a^{13}z^{7}-13a^{13}z^{5}+22a^{13}z^{3}-21a^{13}z+7a^{13}z^{-1}+a^{12}z^{10}+5a^{12}z^{8}-18a^{12}z^{6}+33a^{12}z^{4}-35a^{12}z^{2}-7a^{12}z^{-2}+22a^{12}+6a^{11}z^{9}-11a^{11}z^{7}-8a^{11}z^{5}+46a^{11}z^{3}-45a^{11}z+15a^{11}z^{-1}+a^{10}z^{10}+8a^{10}z^{8}-38a^{10}z^{6}+62a^{10}z^{4}-51a^{10}z^{2}-8a^{10}z^{-2}+28a^{10}+4a^{9}z^{9}-8a^{9}z^{7}-7a^{9}z^{5}+27a^{9}z^{3}-24a^{9}z+8a^{9}z^{-1}+6a^{8}z^{8}-23a^{8}z^{6}+31a^{8}z^{4}-24a^{8}z^{2}-3a^{8}z^{-2}+13a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+2a^{7}z^{3}+a^{6}z^{6}-3a^{6}z^{4}+a^{6}z^{2}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

3

1

-2

-9

4

-11

4

3

-1

-13

10

4

6

-15

7

0

-17

8

7

1

-19

4

7

3

-21

5

8

-3

-23

1

4

3

-25

1

5

-4

-27

1

-29

1

-1

Integral Khovanov Homology

(db, data source)

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