L11a451

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(2)^{3}+t(1)t(3)t(2)^{3}-2t(3)t(2)^{3}+t(2)^{3}+t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-2t(3)^{3}t(2)-3t(1)t(3)^{2}t(2)+3t(3)^{2}t(2)+2t(1)t(3)t(2)-t(3)t(2)-t(1)t(3)^{3}+t(3)^{3}+2t(1)t(3)^{2}-t(3)^{2}}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-4q^{5}+7q^{4}+q^{-4}-8q^{3}-2q^{-3}+11q^{2}+5q^{-2}-10q-7q^{-1}+9$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}-2z^{2}a^{2}+z^{4}-z^{2}+z^{-2}+z^{4}a^{-2}-2z^{2}a^{-2}-2a^{-2}z^{-2}-3a^{-2}+z^{4}a^{-4}+z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-z^{2}a^{-6}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+5z^{9}a^{-3}+3z^{9}a^{-5}+2z^{8}a^{-2}+2z^{8}a^{-4}+3z^{8}a^{-6}+3z^{8}+3az^{7}-z^{7}a^{-1}-17z^{7}a^{-3}-12z^{7}a^{-5}+z^{7}a^{-7}+3a^{2}z^{6}-16z^{6}a^{-2}-22z^{6}a^{-4}-14z^{6}a^{-6}-5z^{6}+2a^{3}z^{5}-2az^{5}-7z^{5}a^{-1}+12z^{5}a^{-3}+11z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-3a^{2}z^{4}+29z^{4}a^{-2}+35z^{4}a^{-4}+18z^{4}a^{-6}+8z^{4}-2a^{3}z^{3}+14z^{3}a^{-1}+8z^{3}a^{-3}-z^{3}a^{-5}+3z^{3}a^{-7}-2a^{4}z^{2}+a^{2}z^{2}-28z^{2}a^{-2}-23z^{2}a^{-4}-7z^{2}a^{-6}-9z^{2}-11za^{-1}-11za^{-3}+a^{4}+13a^{-2}+8a^{-4}+5+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

15

1

-1

13

2

11

2

1

-1

9

5

2

3

7

5

4

-1

5

6

3

4

5

1

5

6

-1

3

5

2

-3

2

4

-2

-5

3

-7

1

2

-1

-9

1

Integral Khovanov Homology

(db, data source)

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