L11n299

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,19,14,18 X_17,11,18,22 X_7,17,8,16 X_21,8,22,9 X_9,14,10,15 X_15,20,16,21 X_19,5,20,10 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -5, 6, -7, 9}, {11, -2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{2}t(3)^{4}+t(2)t(3)^{4}+2t(2)^{2}t(3)^{3}-t(2)t(3)^{3}-t(3)^{3}-t(1)t(2)^{2}t(3)^{2}-t(2)^{2}t(3)^{2}+t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(3)^{2}+t(1)t(2)^{2}t(3)-2t(1)t(3)+t(1)t(2)t(3)+t(1)-t(1)t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{-7}+3q^{-6}-3q^{-5}+4q^{-4}+q^{3}-2q^{-3}-2q^{2}+3q^{-2}+q$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}+a^{6}z^{-2}-a^{6}+2z^{4}a^{4}+6z^{2}a^{4}-2a^{4}z^{-2}+a^{4}-z^{6}a^{2}-5z^{4}a^{2}-6z^{2}a^{2}+a^{2}z^{-2}-a^{2}+z^{2}+1+z^{2}a^{-2}$ (db)
Kauffman polynomial	$2a^{5}z^{9}+2a^{3}z^{9}+3a^{6}z^{8}+6a^{4}z^{8}+4a^{2}z^{8}+z^{8}+a^{7}z^{7}-8a^{5}z^{7}-9a^{3}z^{7}+2az^{7}+2z^{7}a^{-1}-15a^{6}z^{6}-35a^{4}z^{6}-27a^{2}z^{6}+z^{6}a^{-2}-6z^{6}-4a^{7}z^{5}+a^{5}z^{5}-15az^{5}-10z^{5}a^{-1}+19a^{6}z^{4}+55a^{4}z^{4}+48a^{2}z^{4}-4z^{4}a^{-2}+8z^{4}+3a^{7}z^{3}+10a^{5}z^{3}+18a^{3}z^{3}+21az^{3}+10z^{3}a^{-1}-8a^{6}z^{2}-29a^{4}z^{2}-30a^{2}z^{2}+2z^{2}a^{-2}-7z^{2}-a^{7}z-3a^{5}z-7a^{3}z-7az-2za^{-1}+3a^{4}+4a^{2}+2-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

1

-1

3

1

-1

1

2

1

-1

2

3

1

0

-3

3

-5

3

5

1

-7

2

-9

2

4

1

-11

1

0

-13

2

-15

1

-1

Integral Khovanov Homology

(db, data source)

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