L11a437

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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_14,8,15,7 X_8,14,9,13 X_22,15,13,16 X_20,17,21,18 X_16,21,17,22 X_12,19,5,20 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, {5, -4, 6, -8, 7, -3, 9, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v-1)(w-1)\left(3uv-2u+2v^{2}-3v\right)}{{\sqrt {u}}v^{3/2}{\sqrt {w}}}}$ (db)
Jones polynomial	$-q^{-10}+3q^{-9}-6q^{-8}+9q^{-7}-10q^{-6}+13q^{-5}-12q^{-4}+11q^{-3}-7q^{-2}+q+5q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3a^{8}z^{2}+2a^{8}-2a^{6}z^{4}-2a^{6}z^{2}+a^{6}z^{-2}-2a^{4}z^{4}-2a^{4}z^{2}-2a^{4}z^{-2}-3a^{4}-a^{2}z^{4}+a^{2}z^{-2}+a^{2}+z^{2}+1$ (db)
Kauffman polynomial	$z^{7}a^{11}-4z^{5}a^{11}+5z^{3}a^{11}-2za^{11}+3z^{8}a^{10}-12z^{6}a^{10}+14z^{4}a^{10}-7z^{2}a^{10}+2a^{10}+3z^{9}a^{9}-8z^{7}a^{9}-2z^{5}a^{9}+11z^{3}a^{9}-4za^{9}+z^{10}a^{8}+5z^{8}a^{8}-30z^{6}a^{8}+38z^{4}a^{8}-18z^{2}a^{8}+4a^{8}+6z^{9}a^{7}-17z^{7}a^{7}+10z^{5}a^{7}-z^{3}a^{7}+z^{10}a^{6}+5z^{8}a^{6}-21z^{6}a^{6}+19z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-3a^{6}+3z^{9}a^{5}-5z^{7}a^{5}+6z^{5}a^{5}-10z^{3}a^{5}+8za^{5}-2a^{5}z^{-1}+3z^{8}a^{4}-9z^{4}a^{4}+13z^{2}a^{4}+2a^{4}z^{-2}-8a^{4}+3z^{7}a^{3}-5z^{3}a^{3}+6za^{3}-2a^{3}z^{-1}+3z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}-3a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

-1

4

1

3

-3

5

3

-2

-5

6

2

4

-7

6

5

-1

-9

7

6

1

-11

5

8

3

-13

4

5

-1

-15

2

5

3

-17

1

4

-3

-19

2

-21

1

-1

Integral Khovanov Homology

(db, data source)

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