L11a14

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,10,19,9 X₈₄₉₃ X_14,6,15,5 X_20,16,21,15 X_22,18,5,17 X_16,22,17,21 X_10,20,11,19 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4, 10, -7, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(2v^{4}-4v^{3}+5v^{2}-4v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$22q^{9/2}-22q^{7/2}+17q^{5/2}-13q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+5q^{17/2}-10q^{15/2}+16q^{13/2}-20q^{11/2}+6{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-z^{3}a^{-7}+2za^{-7}+2a^{-7}z^{-1}+z^{7}a^{-5}+2z^{5}a^{-5}-3z^{3}a^{-5}-9za^{-5}-5a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+9za^{-3}+3a^{-3}z^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}+5z^{6}a^{-10}-5z^{4}a^{-10}+a^{-10}+10z^{7}a^{-9}-14z^{5}a^{-9}+3z^{3}a^{-9}+11z^{8}a^{-8}-14z^{6}a^{-8}+z^{4}a^{-8}+7z^{9}a^{-7}-z^{7}a^{-7}-12z^{5}a^{-7}+6z^{3}a^{-7}-3za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+13z^{8}a^{-6}-28z^{6}a^{-6}+12z^{4}a^{-6}+6z^{2}a^{-6}-5a^{-6}+11z^{9}a^{-5}-17z^{7}a^{-5}+2z^{5}a^{-5}+14z^{3}a^{-5}-13za^{-5}+5a^{-5}z^{-1}+2z^{10}a^{-4}+6z^{8}a^{-4}-17z^{6}a^{-4}+9z^{4}a^{-4}+7z^{2}a^{-4}-5a^{-4}+4z^{9}a^{-3}-3z^{7}a^{-3}-10z^{5}a^{-3}+20z^{3}a^{-3}-14za^{-3}+3a^{-3}z^{-1}+4z^{8}a^{-2}-7z^{6}a^{-2}+3z^{2}a^{-2}+3z^{7}a^{-1}-9z^{5}a^{-1}+9z^{3}a^{-1}-4za^{-1}+z^{6}-3z^{4}+2z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

20

1

18

4

-4

16

6

1

5

14

10

4

-6

12

10

6

4

10

12

10

-2

8

10

0

6

7

12

5

4

6

10

-4

2

9

7

0

1

4

-3

-2

2

-4

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$