L11a486

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)^{2}\left(t(3)^{3}+2t(2)t(3)^{2}-2t(3)^{2}-2t(2)t(3)+2t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+10q^{7}-17q^{6}+23q^{5}-25q^{4}+27q^{3}-22q^{2}+17q-9+4q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}+a^{-8}-3z^{4}a^{-6}-5z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+2z^{6}a^{-4}+5z^{4}a^{-4}+4z^{2}a^{-4}-2a^{-4}z^{-2}-a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$2z^{10}a^{-4}+2z^{10}a^{-6}+8z^{9}a^{-3}+15z^{9}a^{-5}+7z^{9}a^{-7}+11z^{8}a^{-2}+25z^{8}a^{-4}+22z^{8}a^{-6}+8z^{8}a^{-8}+8z^{7}a^{-1}+z^{7}a^{-3}-14z^{7}a^{-5}-3z^{7}a^{-7}+4z^{7}a^{-9}-16z^{6}a^{-2}-67z^{6}a^{-4}-65z^{6}a^{-6}-17z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-10z^{5}a^{-1}-22z^{5}a^{-3}-23z^{5}a^{-5}-20z^{5}a^{-7}-8z^{5}a^{-9}+12z^{4}a^{-2}+63z^{4}a^{-4}+62z^{4}a^{-6}+14z^{4}a^{-8}-2z^{4}a^{-10}-5z^{4}-az^{3}+5z^{3}a^{-1}+19z^{3}a^{-3}+27z^{3}a^{-5}+20z^{3}a^{-7}+6z^{3}a^{-9}-4z^{2}a^{-2}-27z^{2}a^{-4}-30z^{2}a^{-6}-8z^{2}a^{-8}+z^{2}a^{-10}+2z^{2}-za^{-1}-za^{-3}-5za^{-5}-7za^{-7}-2za^{-9}-a^{-2}+2a^{-4}+5a^{-6}+3a^{-8}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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

19

1

17

3

-3

15

7

1

6

13

10

3

-7

11

13

7

6

9

14

12

-2

7

13

11

2

5

9

14

5

3

8

13

-5

1

3

11

8

-1

1

6

-5

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$