L11a233

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Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_18,6,19,5 X_20,12,21,11 X_22,16,7,15 X_10,18,11,17 X_16,22,17,21 X_14,10,15,9 X_4,20,5,19 X₂₇₃₈ X_6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 8, -6, 4, -2, 11, -8, 5, -7, 6, -3, 9, -4, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {3u^{2}v^{3}-7u^{2}v^{2}+6u^{2}v-2u^{2}+2uv^{4}-11uv^{3}+17uv^{2}-11uv+2u-2v^{4}+6v^{3}-7v^{2}+3v}{uv^{2}}}$ (db)
Jones polynomial	$25q^{9/2}-23q^{7/2}+16q^{5/2}-10q^{3/2}+q^{21/2}-4q^{19/2}+10q^{17/2}-16q^{15/2}+22q^{13/2}-26q^{11/2}+4{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}+z^{3}a^{-7}-2a^{-7}z^{-1}-3z^{5}a^{-5}-5z^{3}a^{-5}-3za^{-5}-z^{5}a^{-3}+2z^{3}a^{-3}+4za^{-3}+a^{-3}z^{-1}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-2z^{4}a^{-12}+z^{2}a^{-12}+4z^{7}a^{-11}-8z^{5}a^{-11}+4z^{3}a^{-11}+8z^{8}a^{-10}-18z^{6}a^{-10}+14z^{4}a^{-10}-7z^{2}a^{-10}+2a^{-10}+8z^{9}a^{-9}-13z^{7}a^{-9}+5z^{5}a^{-9}-4z^{3}a^{-9}+2za^{-9}-a^{-9}z^{-1}+3z^{10}a^{-8}+13z^{8}a^{-8}-43z^{6}a^{-8}+39z^{4}a^{-8}-19z^{2}a^{-8}+5a^{-8}+17z^{9}a^{-7}-31z^{7}a^{-7}+14z^{5}a^{-7}-z^{3}a^{-7}+2za^{-7}-2a^{-7}z^{-1}+3z^{10}a^{-6}+17z^{8}a^{-6}-48z^{6}a^{-6}+41z^{4}a^{-6}-14z^{2}a^{-6}+3a^{-6}+9z^{9}a^{-5}-5z^{7}a^{-5}-14z^{5}a^{-5}+18z^{3}a^{-5}-5za^{-5}+12z^{8}a^{-4}-20z^{6}a^{-4}+14z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}+9z^{7}a^{-3}-14z^{5}a^{-3}+10z^{3}a^{-3}-5za^{-3}+a^{-3}z^{-1}+4z^{6}a^{-2}-4z^{4}a^{-2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

3

18

7

1

-6

16

9

3

6

14

13

7

-6

12

13

9

4

10

12

13

1

8

11

13

-2

6

13

7

4

10

-6

2

1

7

6

0

3

-3

-2

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-2$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=4$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=5$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=6$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=9$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$