L11a431

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Link Presentations

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)(w-1)\left(v^{2}w^{2}-v^{2}w-vw^{2}+vw-v-w+1\right)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+7q^{5}-11q^{4}-q^{-4}+16q^{3}+4q^{-3}-17q^{2}-7q^{-2}+18q+12q^{-1}-14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-8z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}+4z^{2}+a^{2}+3a^{-2}-a^{-4}-3+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+5az^{9}+11z^{9}a^{-1}+6z^{9}a^{-3}+4a^{2}z^{8}+9z^{8}a^{-2}+8z^{8}a^{-4}+5z^{8}+a^{3}z^{7}-16az^{7}-31z^{7}a^{-1}-6z^{7}a^{-3}+8z^{7}a^{-5}-15a^{2}z^{6}-36z^{6}a^{-2}-8z^{6}a^{-4}+7z^{6}a^{-6}-36z^{6}-3a^{3}z^{5}+11az^{5}+19z^{5}a^{-1}-5z^{5}a^{-3}-6z^{5}a^{-5}+4z^{5}a^{-7}+16a^{2}z^{4}+28z^{4}a^{-2}-4z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+40z^{4}+2a^{3}z^{3}+2z^{3}a^{-3}-3z^{3}a^{-5}-3z^{3}a^{-7}-5a^{2}z^{2}+z^{2}a^{-2}+5z^{2}a^{-4}+2z^{2}a^{-6}-7z^{2}+az+3za^{-1}+3za^{-3}+za^{-5}-2a^{2}-6a^{-2}-2a^{-4}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

-3

11

4

1

3

9

7

3

-4

7

9

4

5

8

7

-1

3

10

9

1

8

12

4

-1

4

6

-2

-3

3

8

5

-5

1

4

-3

-7

3

-9

1

-1

Integral Khovanov Homology

(db, data source)

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