L11a542

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Link Presentations

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)(x-1)(2vwx+v(-w)-vx+v-wx+w+x-2)}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$25q^{9/2}-27q^{7/2}+20q^{5/2}-17q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+5q^{17/2}-11q^{15/2}+17q^{13/2}-24q^{11/2}+8{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+3z^{5}a^{-3}+2z^{5}a^{-5}-z^{5}a^{-7}-2z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-z^{3}a^{-7}+2za^{-3}-3za^{-5}+za^{-7}+2a^{-1}z^{-1}-4a^{-3}z^{-1}+2a^{-5}z^{-1}+a^{-1}z^{-3}-3a^{-3}z^{-3}+3a^{-5}z^{-3}-a^{-7}z^{-3}$ (db)
Kauffman polynomial	$z^{5}a^{-11}+5z^{6}a^{-10}-4z^{4}a^{-10}+11z^{7}a^{-9}-15z^{5}a^{-9}+6z^{3}a^{-9}+13z^{8}a^{-8}-16z^{6}a^{-8}+3z^{4}a^{-8}+z^{2}a^{-8}+8z^{9}a^{-7}+6z^{7}a^{-7}-31z^{5}a^{-7}+20z^{3}a^{-7}+a^{-7}z^{-3}-3za^{-7}-2a^{-7}z^{-1}+2z^{10}a^{-6}+22z^{8}a^{-6}-43z^{6}a^{-6}+14z^{4}a^{-6}+3z^{2}a^{-6}-3a^{-6}z^{-2}+4a^{-6}+13z^{9}a^{-5}-5z^{7}a^{-5}-35z^{5}a^{-5}+36z^{3}a^{-5}+3a^{-5}z^{-3}-11za^{-5}-3a^{-5}z^{-1}+2z^{10}a^{-4}+15z^{8}a^{-4}-32z^{6}a^{-4}+9z^{4}a^{-4}+4z^{2}a^{-4}-6a^{-4}z^{-2}+7a^{-4}+5z^{9}a^{-3}+4z^{7}a^{-3}-30z^{5}a^{-3}+32z^{3}a^{-3}+3a^{-3}z^{-3}-11za^{-3}-3a^{-3}z^{-1}+6z^{8}a^{-2}-9z^{6}a^{-2}+3z^{2}a^{-2}-3a^{-2}z^{-2}+4a^{-2}+4z^{7}a^{-1}-10z^{5}a^{-1}+10z^{3}a^{-1}+a^{-1}z^{-3}-3za^{-1}-2a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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

20

1

18

4

-4

16

7

1

6

14

10

4

-6

12

14

7

10

13

12

-1

8

14

12

2

6

10

17

7

4

7

10

-3

2

3

12

9

0

1

5

-4

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{14}$
$r=3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$