L11a114

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$15q^{9/2}-15q^{7/2}+10q^{5/2}-7q^{3/2}+q^{21/2}-2q^{19/2}+5q^{17/2}-9q^{15/2}+13q^{13/2}-15q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+2za^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}-2z^{3}a^{-7}-3za^{-7}-a^{-7}z^{-1}-2z^{5}a^{-5}-4z^{3}a^{-5}-3za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}+3za^{-3}+2a^{-3}z^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-4z^{4}a^{-12}+4z^{2}a^{-12}+2z^{7}a^{-11}-6z^{5}a^{-11}+4z^{3}a^{-11}+3z^{8}a^{-10}-9z^{6}a^{-10}+12z^{4}a^{-10}-12z^{2}a^{-10}+4a^{-10}+3z^{9}a^{-9}-8z^{7}a^{-9}+12z^{5}a^{-9}-11z^{3}a^{-9}+2za^{-9}-a^{-9}z^{-1}+z^{10}a^{-8}+5z^{8}a^{-8}-22z^{6}a^{-8}+39z^{4}a^{-8}-34z^{2}a^{-8}+9a^{-8}+6z^{9}a^{-7}-12z^{7}a^{-7}+8z^{5}a^{-7}+3z^{3}a^{-7}-za^{-7}-a^{-7}z^{-1}+z^{10}a^{-6}+7z^{8}a^{-6}-21z^{6}a^{-6}+27z^{4}a^{-6}-15z^{2}a^{-6}+4a^{-6}+3z^{9}a^{-5}+3z^{7}a^{-5}-20z^{5}a^{-5}+27z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+5z^{8}a^{-4}-6z^{6}a^{-4}-z^{4}a^{-4}+4z^{2}a^{-4}-2a^{-4}+5z^{7}a^{-3}-9z^{5}a^{-3}+7z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}+3z^{6}a^{-2}-5z^{4}a^{-2}+z^{2}a^{-2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

1

18

4

1

-3

16

5

1

4

14

8

4

-4

12

7

5

2

10

8

0

8

7

0

6

3

8

5

4

7

-3

2

1

5

4

0

2

-2

1

Integral Khovanov Homology

(db, data source)

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