L11a536

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a536 at Knotilus!
	[edit L11a536 Quick Notes]

[edit L11a536 Further Notes and Views]

Link Presentations

[edit Notes on L11a536's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw^{2}x-uvw^{2}+uvwx^{2}-3uvwx+2uvw-uvx^{2}+2uvx-2uw^{2}x+uw^{2}-2uwx^{2}+4uwx-2uw+2ux^{2}-2ux-2vw^{2}x+2vw^{2}-2vwx^{2}+4vwx-2vw+vx^{2}-2vx+2w^{2}x-w^{2}+2wx^{2}-3wx+w-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+7q^{5/2}-12q^{3/2}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}+a^{5}z^{-3}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}-3a^{3}z^{-3}+z^{3}a^{-3}-4a^{3}z^{-1}-az^{5}-z^{5}a^{-1}+2az^{3}+3az^{-3}-z^{3}a^{-1}-a^{-1}z^{-3}+6az+7az^{-1}-3za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+2a^{6}z^{6}-3a^{6}z^{4}+a^{6}+3a^{5}z^{7}-3a^{5}z^{5}+a^{5}z^{3}-a^{5}z^{-3}-3a^{5}z+3a^{5}z^{-1}+3a^{4}z^{8}+a^{4}z^{6}+z^{6}a^{-4}-10a^{4}z^{4}-2z^{4}a^{-4}+16a^{4}z^{2}+3a^{4}z^{-2}-11a^{4}+2a^{3}z^{9}+6a^{3}z^{7}+4z^{7}a^{-3}-19a^{3}z^{5}-11z^{5}a^{-3}+26a^{3}z^{3}+7z^{3}a^{-3}-3a^{3}z^{-3}-21a^{3}z+12a^{3}z^{-1}+a^{2}z^{10}+5a^{2}z^{8}+6z^{8}a^{-2}-3a^{2}z^{6}-16z^{6}a^{-2}-17a^{2}z^{4}+10z^{4}a^{-2}+33a^{2}z^{2}+6a^{2}z^{-2}-24a^{2}+6az^{9}+4z^{9}a^{-1}-5az^{7}-4z^{7}a^{-1}-13az^{5}-9z^{5}a^{-1}+29az^{3}+14z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-28az-13za^{-1}+14az^{-1}+6a^{-1}z^{-1}+z^{10}+8z^{8}-19z^{6}+2z^{4}+17z^{2}+3z^{-2}-13$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

4

1

-3

4

8

3

5

2

9

7

-2

0

9

5

4

-2

7

10

3

-4

8

0

-6

2

8

6

-8

4

7

-3

-10

1

5

4

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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