L11a407

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-4t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-t(2)^{2}-3t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-4t(1)t(2)+7t(1)t(3)t(2)-7t(3)t(2)+3t(2)+t(1)t(3)^{2}-t(3)^{2}+4t(1)-4t(1)t(3)+3t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+13q^{5}-17q^{4}+19q^{3}-18q^{2}+16q-10+7q^{-1}-2q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}-2z^{4}-4z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+6z^{8}a^{-2}+10z^{8}a^{-4}+7z^{8}a^{-6}+3z^{8}+2az^{7}+5z^{7}a^{-1}+4z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-7z^{6}a^{-2}-13z^{6}a^{-4}-7z^{6}a^{-6}+4z^{6}a^{-8}-4z^{6}-4az^{5}-19z^{5}a^{-1}-17z^{5}a^{-3}-14z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}-8z^{4}a^{-2}-5z^{4}a^{-4}-2z^{4}a^{-6}-6z^{4}a^{-8}-3z^{4}+14z^{3}a^{-1}+16z^{3}a^{-3}+8z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}+9z^{2}a^{-2}+9z^{2}a^{-4}+4z^{2}a^{-6}+2z^{2}a^{-8}+8z^{2}+4az-6za^{-3}-2za^{-5}-4a^{2}-3a^{-2}-a^{-4}-a^{-6}-6-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

3

13

5

1

-4

11

8

3

5

9

5

-4

7

10

8

2

5

10

11

1

3

6

8

-2

1

5

11

6

-1

2

5

-3

5

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

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