L11n107

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(3t(2)^{2}-4t(2)+3\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$12q^{9/2}-11q^{7/2}+6q^{5/2}-3q^{3/2}+q^{21/2}-3q^{19/2}+7q^{17/2}-10q^{15/2}+13q^{13/2}-14q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-5}-z^{5}a^{-7}+3z^{3}a^{-3}-5z^{3}a^{-5}-z^{3}a^{-7}+z^{3}a^{-9}+5za^{-3}-5za^{-5}-za^{-7}+za^{-9}+2a^{-3}z^{-1}-2a^{-5}z^{-1}-a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-3z^{4}a^{-12}+3z^{2}a^{-12}-a^{-12}+3z^{7}a^{-11}-8z^{5}a^{-11}+6z^{3}a^{-11}-za^{-11}+4z^{8}a^{-10}-8z^{6}a^{-10}+z^{4}a^{-10}+2z^{2}a^{-10}+2z^{9}a^{-9}+4z^{7}a^{-9}-19z^{5}a^{-9}+12z^{3}a^{-9}-2za^{-9}-a^{-9}z^{-1}+10z^{8}a^{-8}-21z^{6}a^{-8}+11z^{4}a^{-8}-6z^{2}a^{-8}+3a^{-8}+2z^{9}a^{-7}+8z^{7}a^{-7}-25z^{5}a^{-7}+19z^{3}a^{-7}-4za^{-7}-a^{-7}z^{-1}+6z^{8}a^{-6}-9z^{6}a^{-6}+7z^{4}a^{-6}-2z^{2}a^{-6}+7z^{7}a^{-5}-14z^{5}a^{-5}+19z^{3}a^{-5}-10za^{-5}+2a^{-5}z^{-1}+3z^{6}a^{-4}+3z^{2}a^{-4}-3a^{-4}+6z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}$ (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		\

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

2

18

5

1

-4

16

5

2

3

14

8

5

-3

12

6

5

1

10

6

8

2

8

5

6

-1

6

1

6

5

4

2

5

-3

2

3

Integral Khovanov Homology

(db, data source)

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