L11n410

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)\left(2vw^{2}-2vw+v-w^{2}+2w-2\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$1-3q^{-1}+7q^{-2}-9q^{-3}+14q^{-4}-13q^{-5}+13q^{-6}-10q^{-7}+7q^{-8}-3q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+a^{8}z^{4}+3a^{8}z^{2}+4a^{8}z^{-2}+6a^{8}-a^{6}z^{6}-3a^{6}z^{4}-6a^{6}z^{2}-5a^{6}z^{-2}-10a^{6}-a^{4}z^{6}-2a^{4}z^{4}+a^{4}z^{2}+2a^{4}z^{-2}+4a^{4}+a^{2}z^{4}+2a^{2}z^{2}+a^{2}$ (db)
Kauffman polynomial	$6z^{3}a^{11}-5za^{11}+a^{11}z^{-1}+3z^{6}a^{10}+3z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+3a^{10}+8z^{7}a^{9}-19z^{5}a^{9}+29z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+7z^{8}a^{8}-16z^{6}a^{8}+25z^{4}a^{8}-23z^{2}a^{8}-4a^{8}z^{-2}+13a^{8}+2z^{9}a^{7}+9z^{7}a^{7}-33z^{5}a^{7}+41z^{3}a^{7}-30za^{7}+9a^{7}z^{-1}+11z^{8}a^{6}-29z^{6}a^{6}+30z^{4}a^{6}-25z^{2}a^{6}-5a^{6}z^{-2}+16a^{6}+2z^{9}a^{5}+4z^{7}a^{5}-22z^{5}a^{5}+23z^{3}a^{5}-15za^{5}+5a^{5}z^{-1}+4z^{8}a^{4}-9z^{6}a^{4}+5z^{4}a^{4}-5z^{2}a^{4}-2a^{4}z^{-2}+6a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

2

-2

-3

5

1

4

-5

6

4

-2

-7

8

3

5

-9

5

6

1

-11

8

0

-13

4

7

3

-15

3

6

-3

-17

4

-19

3

-3

Integral Khovanov Homology

(db, data source)

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