L11a531

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Link Presentations

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2uvwx^{2}+2uvwx-uvw-uvx^{3}+3uvx^{2}-3uvx+uv-uwx^{3}+3uwx^{2}-3uwx+uw+2ux^{3}-4ux^{2}+4ux-u-vwx^{3}+4vwx^{2}-4vwx+2vw+vx^{3}-3vx^{2}+3vx-v+wx^{3}-3wx^{2}+3wx-w-x^{3}+2x^{2}-2x}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {2}{q^{25/2}}}+{\frac {6}{q^{23/2}}}-{\frac {11}{q^{21/2}}}+{\frac {16}{q^{19/2}}}-{\frac {22}{q^{17/2}}}+{\frac {19}{q^{15/2}}}-{\frac {21}{q^{13/2}}}+{\frac {14}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{15}z^{-3}+5a^{13}z^{-3}+5a^{13}z^{-1}-9a^{11}z^{-3}-10a^{11}z-19a^{11}z^{-1}+10a^{9}z^{3}+7a^{9}z^{-3}+26a^{9}z+23a^{9}z^{-1}-4a^{7}z^{5}-13a^{7}z^{3}-2a^{7}z^{-3}-16a^{7}z-9a^{7}z^{-1}-a^{5}z^{5}-a^{5}z^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+4z^{4}a^{16}-6z^{2}a^{16}-a^{16}z^{-2}+4a^{16}-2z^{7}a^{15}+5z^{5}a^{15}-4z^{3}a^{15}+2za^{15}-2a^{15}z^{-1}+a^{15}z^{-3}-2z^{8}a^{14}-3z^{6}a^{14}+22z^{4}a^{14}-34z^{2}a^{14}-7a^{14}z^{-2}+24a^{14}-2z^{9}a^{13}-5z^{7}a^{13}+19z^{5}a^{13}-22z^{3}a^{13}+16za^{13}-12a^{13}z^{-1}+5a^{13}z^{-3}-z^{10}a^{12}-7z^{8}a^{12}+3z^{6}a^{12}+38z^{4}a^{12}-75z^{2}a^{12}-18a^{12}z^{-2}+58a^{12}-7z^{9}a^{11}-3z^{7}a^{11}+34z^{5}a^{11}-45z^{3}a^{11}+37za^{11}-24a^{11}z^{-1}+9a^{11}z^{-3}-z^{10}a^{10}-15z^{8}a^{10}+19z^{6}a^{10}+29z^{4}a^{10}-73z^{2}a^{10}-19a^{10}z^{-2}+60a^{10}-5z^{9}a^{9}-10z^{7}a^{9}+40z^{5}a^{9}-47z^{3}a^{9}+39za^{9}-23a^{9}z^{-1}+7a^{9}z^{-3}-10z^{8}a^{8}+10z^{6}a^{8}+12z^{4}a^{8}-26z^{2}a^{8}-7a^{8}z^{-2}+23a^{8}-10z^{7}a^{7}+19z^{5}a^{7}-19z^{3}a^{7}+16za^{7}-9a^{7}z^{-1}+2a^{7}z^{-3}-4z^{6}a^{6}+3z^{4}a^{6}-z^{5}a^{5}+z^{3}a^{5}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

4

1

-3

-8

7

-10

7

4

-3

-12

14

7

-14

11

13

2

-16

11

8

3

-18

5

11

6

-20

6

11

-5

-22

1

6

5

-24

1

5

-4

-26

1

-28

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-7$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-6$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-4$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{14}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$