L11a414

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{2}t(3)^{4}-2t(2)^{2}t(3)^{4}-t(1)t(2)t(3)^{4}+t(2)t(3)^{4}-t(1)t(2)^{2}t(3)^{3}+2t(2)^{2}t(3)^{3}-t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+t(2)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-2t(2)t(3)+t(3)+2t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+9q^{6}-11q^{5}+12q^{4}-11q^{3}+10q^{2}-6q+5-q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-11z^{4}a^{-2}+14z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-21z^{2}a^{-2}+18z^{2}a^{-4}-5z^{2}a^{-6}+5z^{2}-17a^{-2}+13a^{-4}-3a^{-6}+7-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+z^{9}a^{-1}+5z^{9}a^{-3}+4z^{9}a^{-5}-z^{8}a^{-2}+6z^{8}a^{-4}+8z^{8}a^{-6}+z^{8}-3z^{7}a^{-1}-16z^{7}a^{-3}-3z^{7}a^{-5}+10z^{7}a^{-7}-13z^{6}a^{-2}-32z^{6}a^{-4}-17z^{6}a^{-6}+9z^{6}a^{-8}-7z^{6}-4z^{5}a^{-1}-z^{5}a^{-3}-23z^{5}a^{-5}-20z^{5}a^{-7}+6z^{5}a^{-9}+36z^{4}a^{-2}+37z^{4}a^{-4}+3z^{4}a^{-6}-13z^{4}a^{-8}+3z^{4}a^{-10}+18z^{4}+20z^{3}a^{-1}+39z^{3}a^{-3}+33z^{3}a^{-5}+9z^{3}a^{-7}-4z^{3}a^{-9}+z^{3}a^{-11}-39z^{2}a^{-2}-20z^{2}a^{-4}+3z^{2}a^{-6}+5z^{2}a^{-8}-21z^{2}-19za^{-1}-35za^{-3}-19za^{-5}-2za^{-7}+za^{-9}+22a^{-2}+13a^{-4}-a^{-8}+11+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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

19

1

-1

17

2

15

4

1

-3

13

5

2

3

11

6

4

-2

9

6

5

1

7

8

1

5

3

4

-1

3

4

8

4

1

2

-1

4

-3

1

0

-5

1

Integral Khovanov Homology

(db, data source)

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