L11n315

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,17,8,16 X_9,20,10,21 X_11,18,12,19 X_19,22,20,11 X_15,9,16,8 X_21,10,22,5 X_17,14,18,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, -4, 8}, {-5, -2, 11, 9, -7, 3, -9, 5, -6, 4, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{2}-t(1)t(3)t(2)^{2}-t(3)t(2)^{2}+t(2)^{2}+t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)-t(1)t(3)t(2)+t(3)t(2)-t(2)-t(1)t(3)^{2}-t(3)^{2}+t(1)t(3)+t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-2q^{-5}+2q^{-4}+q^{3}-q^{-3}-q^{2}+q^{-2}+2q+q^{-1}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+a^{2}+2a^{-2}-z^{4}-4z^{2}-2z^{-2}-4$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+3a^{4}z^{8}+a^{2}z^{8}+a^{7}z^{7}-4a^{5}z^{7}-6a^{3}z^{7}+z^{7}a^{-1}-11a^{6}z^{6}-19a^{4}z^{6}-7a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5a^{7}z^{5}-a^{5}z^{5}+8a^{3}z^{5}-4z^{5}a^{-1}+16a^{6}z^{4}+33a^{4}z^{4}+11a^{2}z^{4}-5z^{4}a^{-2}-11z^{4}+6a^{7}z^{3}+9a^{5}z^{3}-3a^{3}z^{3}-5az^{3}+z^{3}a^{-1}-8a^{6}z^{2}-20a^{4}z^{2}-5a^{2}z^{2}+7z^{2}a^{-2}+14z^{2}-2a^{7}z-4a^{5}z+6az+4za^{-1}+2a^{6}+4a^{4}-2a^{2}-4a^{-2}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

0

3

2

1

2

1

2

-1

1

3

1

-3

2

-5

2

3

1

0

-7

1

-9

1

2

1

0

-11

1

0

-13

1

-15

1

-1

Integral Khovanov Homology

(db, data source)

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