L11a117

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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_22,14,5,13 X_8,22,9,21 X_20,10,21,9 X_18,12,19,11 X_12,18,13,17 X_10,20,11,19 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 6, -9, 7, -8, 4, -2, 11, -3, 8, -7, 9, -6, 5, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(3t(2)^{2}-2t(2)+3\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{21/2}-2q^{19/2}+4q^{17/2}-6q^{15/2}+8q^{13/2}-10q^{11/2}+10q^{9/2}-9q^{7/2}+6q^{5/2}-5q^{3/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-z^{5}a^{-5}-z^{5}a^{-7}+z^{3}a^{-1}-2z^{3}a^{-3}-z^{3}a^{-5}-3z^{3}a^{-7}+z^{3}a^{-9}+2za^{-1}-za^{-3}+za^{-5}-4za^{-7}+2za^{-9}+a^{-1}z^{-1}-a^{-3}z^{-1}+a^{-5}z^{-1}-2a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-5z^{9}a^{-7}-3z^{9}a^{-9}-2z^{8}a^{-4}-z^{8}a^{-8}-3z^{8}a^{-10}-2z^{7}a^{-3}+5z^{7}a^{-5}+22z^{7}a^{-7}+13z^{7}a^{-9}-2z^{7}a^{-11}-2z^{6}a^{-2}+2z^{6}a^{-4}+9z^{6}a^{-6}+18z^{6}a^{-8}+12z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+z^{5}a^{-3}-8z^{5}a^{-5}-38z^{5}a^{-7}-21z^{5}a^{-9}+7z^{5}a^{-11}+4z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}a^{-6}-39z^{4}a^{-8}-15z^{4}a^{-10}+4z^{4}a^{-12}+3z^{3}a^{-1}+6z^{3}a^{-3}+7z^{3}a^{-5}+25z^{3}a^{-7}+17z^{3}a^{-9}-4z^{3}a^{-11}+15z^{2}a^{-6}+29z^{2}a^{-8}+11z^{2}a^{-10}-3z^{2}a^{-12}-3za^{-1}-3za^{-3}-5za^{-5}-10za^{-7}-5za^{-9}-a^{-2}-4a^{-6}-7a^{-8}-3a^{-10}+a^{-1}z^{-1}+a^{-3}z^{-1}+a^{-5}z^{-1}+2a^{-7}z^{-1}+a^{-9}z^{-1}$ (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

3

1

-2

16

3

1

2

14

5

3

-2

12

5

3

2

10

5

0

8

4

5

-1

6

2

5

3

4

3

4

-1

2

1

4

3

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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