L11a13

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-5v^{3}+7v^{2}-5v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+16q^{7/2}-22q^{5/2}+24q^{3/2}-25{\sqrt {q}}+{\frac {21}{\sqrt {q}}}-{\frac {16}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+4az^{3}-5z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+3az-4za^{-1}+3za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-6az^{9}-13z^{9}a^{-1}-7z^{9}a^{-3}-7a^{2}z^{8}-19z^{8}a^{-2}-10z^{8}a^{-4}-16z^{8}-4a^{3}z^{7}+2az^{7}+11z^{7}a^{-1}-3z^{7}a^{-3}-8z^{7}a^{-5}-a^{4}z^{6}+14a^{2}z^{6}+43z^{6}a^{-2}+12z^{6}a^{-4}-4z^{6}a^{-6}+42z^{6}+9a^{3}z^{5}+18az^{5}+20z^{5}a^{-1}+23z^{5}a^{-3}+11z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-8a^{2}z^{4}-27z^{4}a^{-2}-4z^{4}a^{-4}+5z^{4}a^{-6}-28z^{4}-7a^{3}z^{3}-18az^{3}-23z^{3}a^{-1}-20z^{3}a^{-3}-7z^{3}a^{-5}+z^{3}a^{-7}-a^{4}z^{2}+a^{2}z^{2}+5z^{2}a^{-2}-z^{2}a^{-4}-2z^{2}a^{-6}+6z^{2}+2a^{3}z+6az+8za^{-1}+6za^{-3}+2za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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

3

-3

10

6

1

5

8

10

3

-7

6

12

6

4

12

10

-2

2

13

12

1

0

10

14

4

-2

6

11

-5

-4

3

10

7

-6

1

6

-5

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$