L11a49

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,9,17,10 X_4,21,1,22 X_14,12,15,11 X_10,4,11,3 X_12,5,13,6 X_22,13,5,14 X_2,16,3,15 X_20,18,21,17 X_18,8,19,7 X_8,20,9,19
Gauss code	{1, -8, 5, -3}, {6, -1, 10, -11, 2, -5, 4, -6, 7, -4, 8, -2, 9, -10, 11, -9, 3, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{3}\left(v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-11q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {4}{q^{7/2}}}-23q^{5/2}+{\frac {9}{q^{5/2}}}+26q^{3/2}-{\frac {16}{q^{3/2}}}-q^{13/2}+5q^{11/2}-26{\sqrt {q}}+{\frac {21}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+2z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-a^{3}z+za^{-3}-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+4az^{3}-5z^{3}a^{-1}+3az-3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-6az^{9}-14z^{9}a^{-1}-8z^{9}a^{-3}-7a^{2}z^{8}-23z^{8}a^{-2}-13z^{8}a^{-4}-17z^{8}-4a^{3}z^{7}+2az^{7}+12z^{7}a^{-1}-5z^{7}a^{-3}-11z^{7}a^{-5}-a^{4}z^{6}+14a^{2}z^{6}+51z^{6}a^{-2}+17z^{6}a^{-4}-5z^{6}a^{-6}+44z^{6}+9a^{3}z^{5}+17az^{5}+20z^{5}a^{-1}+28z^{5}a^{-3}+15z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-8a^{2}z^{4}-29z^{4}a^{-2}-5z^{4}a^{-4}+4z^{4}a^{-6}-30z^{4}-7a^{3}z^{3}-18az^{3}-22z^{3}a^{-1}-16z^{3}a^{-3}-5z^{3}a^{-5}-a^{4}z^{2}+a^{2}z^{2}+3z^{2}a^{-2}+5z^{2}+2a^{3}z+6az+6za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

4

-4

10

7

1

6

8

10

4

-6

6

13

7

6

4

13

10

-3

2

13

0

10

15

5

-2

6

11

-5

-4

3

10

7

-6

1

6

-5

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$