L11a454

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-t(3)^{2}t(2)^{3}+t(1)t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-2t(3)^{3}t(2)^{2}-4t(1)t(3)^{2}t(2)^{2}+7t(3)^{2}t(2)^{2}-3t(1)t(2)^{2}+6t(1)t(3)t(2)^{2}-6t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+3t(3)^{3}t(2)+6t(1)t(3)^{2}t(2)-6t(3)^{2}t(2)+2t(1)t(2)-7t(1)t(3)t(2)+4t(3)t(2)-t(2)+t(1)t(3)^{3}-t(3)^{3}-2t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+4q^{4}+8q^{-4}-10q^{3}-14q^{-3}+17q^{2}+21q^{-2}-21q-23q^{-1}+25$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3a^{4}z^{2}+a^{4}z^{-2}-z^{2}a^{-4}-a^{4}+3a^{2}z^{4}+2z^{4}a^{-2}+a^{2}z^{2}-2a^{2}z^{-2}-3a^{2}-a^{-2}-z^{6}+2z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+13az^{9}+8z^{9}a^{-1}+5a^{4}z^{8}+13a^{2}z^{8}+12z^{8}a^{-2}+20z^{8}+3a^{5}z^{7}-3a^{3}z^{7}-16az^{7}-z^{7}a^{-1}+9z^{7}a^{-3}+a^{6}z^{6}-9a^{4}z^{6}-39a^{2}z^{6}-20z^{6}a^{-2}+4z^{6}a^{-4}-53z^{6}-7a^{5}z^{5}-10a^{3}z^{5}-8az^{5}-19z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+5a^{4}z^{4}+41a^{2}z^{4}+16z^{4}a^{-2}-4z^{4}a^{-4}+53z^{4}+5a^{5}z^{3}+14a^{3}z^{3}+19az^{3}+19z^{3}a^{-1}+8z^{3}a^{-3}-z^{3}a^{-5}+3a^{6}z^{2}-3a^{4}z^{2}-27a^{2}z^{2}-10z^{2}a^{-2}+z^{2}a^{-4}-32z^{2}-a^{5}z-8a^{3}z-12az-7za^{-1}-2za^{-3}-a^{6}+3a^{4}+11a^{2}+3a^{-2}+11+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

1

-6

5

10

3

7

3

11

7

-4

1

14

10

4

-1

12

14

2

-3

9

11

-2

-5

5

12

7

-7

3

9

-6

-9

1

6

5

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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