L11a470

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)\left(2uv^{2}w^{2}-2uv^{2}w-3uvw^{2}+3uvw-uv+uw^{2}-2uw+u+v^{2}w^{3}-2v^{2}w^{2}+v^{2}w-vw^{3}+3vw^{2}-3vw-2w^{2}+2w\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$1-3q^{-1}+8q^{-2}-12q^{-3}+17q^{-4}-19q^{-5}+20q^{-6}-16q^{-7}+13q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-2a^{10}+3z^{4}a^{8}+8z^{2}a^{8}+a^{8}z^{-2}+6a^{8}-2z^{6}a^{6}-7z^{4}a^{6}-10z^{2}a^{6}-2a^{6}z^{-2}-8a^{6}-z^{6}a^{4}-z^{4}a^{4}+3z^{2}a^{4}+a^{4}z^{-2}+3a^{4}+z^{4}a^{2}+2z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-2a^{13}z^{3}+a^{13}z+3a^{12}z^{6}-5a^{12}z^{4}+3a^{12}z^{2}-a^{12}+5a^{11}z^{7}-6a^{11}z^{5}+a^{11}z^{3}+a^{11}z+6a^{10}z^{8}-7a^{10}z^{6}+4a^{10}z^{4}-2a^{10}z^{2}+4a^{9}z^{9}+3a^{9}z^{7}-15a^{9}z^{5}+14a^{9}z^{3}-3a^{9}z+a^{8}z^{10}+14a^{8}z^{8}-38a^{8}z^{6}+46a^{8}z^{4}-31a^{8}z^{2}-a^{8}z^{-2}+10a^{8}+8a^{7}z^{9}-7a^{7}z^{7}-12a^{7}z^{5}+19a^{7}z^{3}-10a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+13a^{6}z^{8}-40a^{6}z^{6}+48a^{6}z^{4}-36a^{6}z^{2}-2a^{6}z^{-2}+14a^{6}+4a^{5}z^{9}-2a^{5}z^{7}-11a^{5}z^{5}+12a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-11a^{4}z^{6}+8a^{4}z^{4}-7a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+3a^{3}z^{7}-7a^{3}z^{5}+4a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+3a^{2}z^{2}-a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

2

-2

-3

6

1

5

-5

7

3

-4

-7

10

5

-9

10

8

-2

-11

10

9

1

-13

7

11

4

-15

6

9

-3

-17

2

8

6

-19

1

5

-4

-21

2

-23

1

-1

Integral Khovanov Homology

(db, data source)

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