L11a423

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,10,21,9 X_16,8,17,7 X_18,14,19,13 X_14,20,15,19 X_22,16,11,15 X_10,18,5,17 X_8,22,9,21 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {11, -2, 5, -6, 7, -4, 8, -5, 6, -3, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(3)^{4}+t(1)t(2)t(3)^{4}-2t(2)t(3)^{4}+t(3)^{4}+t(1)t(2)^{2}t(3)^{3}-3t(2)^{2}t(3)^{3}+3t(1)t(3)^{3}-4t(1)t(2)t(3)^{3}+6t(2)t(3)^{3}-2t(3)^{3}-2t(1)t(2)^{2}t(3)^{2}+4t(2)^{2}t(3)^{2}-4t(1)t(3)^{2}+7t(1)t(2)t(3)^{2}-7t(2)t(3)^{2}+2t(3)^{2}+2t(1)t(2)^{2}t(3)-3t(2)^{2}t(3)+3t(1)t(3)-6t(1)t(2)t(3)+4t(2)t(3)-t(3)-t(1)t(2)^{2}+t(2)^{2}+2t(1)t(2)-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-10q^{6}+16q^{5}-21q^{4}+25q^{3}+q^{-3}-23q^{2}-3q^{-2}+21q+9q^{-1}-14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+6z^{2}a^{-4}+4a^{-4}z^{-2}+9a^{-4}+z^{6}a^{-2}-z^{4}a^{-2}+a^{2}z^{2}-8z^{2}a^{-2}-5a^{-2}z^{-2}+a^{2}-11a^{-2}-2z^{4}-z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-4z^{4}a^{-8}+9z^{7}a^{-7}-14z^{5}a^{-7}+9z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+12z^{8}a^{-6}-22z^{6}a^{-6}+19z^{4}a^{-6}-9z^{2}a^{-6}-a^{-6}z^{-2}+3a^{-6}+8z^{9}a^{-5}-2z^{7}a^{-5}-20z^{5}a^{-5}+29z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+2z^{10}a^{-4}+20z^{8}a^{-4}-54z^{6}a^{-4}+52z^{4}a^{-4}-29z^{2}a^{-4}-4a^{-4}z^{-2}+15a^{-4}+13z^{9}a^{-3}-14z^{7}a^{-3}-21z^{5}a^{-3}+45z^{3}a^{-3}-33za^{-3}+9a^{-3}z^{-1}+2z^{10}a^{-2}+14z^{8}a^{-2}+a^{2}z^{6}-43z^{6}a^{-2}-3a^{2}z^{4}+48z^{4}a^{-2}+3a^{2}z^{2}-38z^{2}a^{-2}-5a^{-2}z^{-2}-a^{2}+20a^{-2}+5z^{9}a^{-1}+3az^{7}-6az^{5}-22z^{5}a^{-1}+3az^{3}+29z^{3}a^{-1}-18za^{-1}+5a^{-1}z^{-1}+6z^{8}-14z^{6}+16z^{4}-15z^{2}-2z^{-2}+8$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

3

13

7

1

-6

11

9

3

6

9

12

7

-5

7

13

9

4

5

12

14

2

3

9

11

-2

1

6

13

7

-1

3

8

-5

-3

6

-5

1

3

-2

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$