L11a467

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)\left(-t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-3t(1)t(2)t(3)^{2}+5t(2)t(3)^{2}-2t(3)^{2}-2t(1)t(2)^{2}t(3)+2t(2)^{2}t(3)-2t(1)t(3)+5t(1)t(2)t(3)-3t(2)t(3)+t(3)+t(1)t(2)^{2}-t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+4q^{4}+8q^{-4}-9q^{3}-13q^{-3}+15q^{2}+19q^{-2}-19q-21q^{-1}+23$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3a^{4}z^{2}+a^{4}z^{-2}-z^{2}a^{-4}-a^{4}+3a^{2}z^{4}+2z^{4}a^{-2}+2a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}-2a^{2}-z^{6}-z^{4}-z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-7a^{5}z^{5}+z^{5}a^{-5}+5a^{5}z^{3}-z^{3}a^{-5}-a^{5}z+5a^{4}z^{8}-10a^{4}z^{6}+4z^{6}a^{-4}+7a^{4}z^{4}-5z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}-a^{4}z^{-2}+4a^{4}+4a^{3}z^{9}+a^{3}z^{7}+8z^{7}a^{-3}-18a^{3}z^{5}-12z^{5}a^{-3}+20a^{3}z^{3}+7z^{3}a^{-3}-10a^{3}z-2za^{-3}+2a^{3}z^{-1}+a^{2}z^{10}+16a^{2}z^{8}+9z^{8}a^{-2}-44a^{2}z^{6}-11z^{6}a^{-2}+46a^{2}z^{4}+5z^{4}a^{-2}-30a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}+12a^{2}+2a^{-2}+9az^{9}+5z^{9}a^{-1}-az^{7}+9z^{7}a^{-1}-31az^{5}-33z^{5}a^{-1}+34az^{3}+27z^{3}a^{-1}-14az-7za^{-1}+2az^{-1}+z^{10}+20z^{8}-48z^{6}+46z^{4}-28z^{2}-z^{-2}+10$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

6

1

-5

5

9

3

6

3

11

7

-4

1

12

8

4

-1

10

12

2

-3

9

11

-2

-5

5

11

6

-7

3

8

-5

-9

1

6

5

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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