L11a473

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)\left(v^{2}w^{3}-v^{2}w^{2}+v^{2}w-vw^{3}+3vw^{2}-3vw+v-w^{2}+w-1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+12q^{2}-15q+19-17q^{-1}+16q^{-2}-11q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+6z^{6}+a^{4}z^{4}-9a^{2}z^{4}-4z^{4}a^{-2}+15z^{4}+3a^{4}z^{2}-15a^{2}z^{2}-6z^{2}a^{-2}+18z^{2}+3a^{4}-10a^{2}-3a^{-2}+10+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+8az^{9}+5z^{9}a^{-1}+4a^{4}z^{8}+10a^{2}z^{8}+8z^{8}a^{-2}+14z^{8}+3a^{5}z^{7}+a^{3}z^{7}-13az^{7}-5z^{7}a^{-1}+6z^{7}a^{-3}+a^{6}z^{6}-7a^{4}z^{6}-32a^{2}z^{6}-22z^{6}a^{-2}+3z^{6}a^{-4}-49z^{6}-8a^{5}z^{5}-15a^{3}z^{5}-7z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+40a^{2}z^{4}+30z^{4}a^{-2}-5z^{4}a^{-4}+72z^{4}+6a^{5}z^{3}+13a^{3}z^{3}+16az^{3}+21z^{3}a^{-1}+10z^{3}a^{-3}-2z^{3}a^{-5}+3a^{6}z^{2}-33a^{2}z^{2}-18z^{2}a^{-2}-48z^{2}-a^{5}z-7a^{3}z-13az-10za^{-1}-3za^{-3}-a^{6}+3a^{4}+13a^{2}+5a^{-2}+15+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

5

1

-4

5

7

2

5

3

8

5

-3

1

11

7

4

-1

8

10

2

-3

8

9

-1

-5

5

10

5

-7

2

6

-4

-9

1

5

4

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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