L11n313

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n313 at Knotilus!
	[edit L11n313 Quick Notes]

[edit L11n313 Further Notes and Views]

Link Presentations

[edit Notes on L11n313's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)t(3)^{4}+t(1)t(2)^{2}t(3)^{3}+t(2)^{2}t(3)^{3}-t(1)t(3)^{3}-t(1)t(2)t(3)^{3}-t(2)t(3)^{3}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}+t(2)^{2}t(3)-t(1)t(3)+t(1)t(2)t(3)+t(2)t(3)-t(3)-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{-3}-q^{-4}+q^{-5}+q^{-6}+3q^{-8}-2q^{-9}+3q^{-10}-3q^{-11}+2q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+5a^{12}-a^{10}z^{4}-9a^{10}z^{2}-5a^{10}z^{-2}-13a^{10}+a^{8}z^{6}+6a^{8}z^{4}+9a^{8}z^{2}+2a^{8}z^{-2}+7a^{8}+a^{6}z^{6}+5a^{6}z^{4}+5a^{6}z^{2}+a^{6}$ (db)
Kauffman polynomial	$z^{7}a^{15}-5z^{5}a^{15}+7z^{3}a^{15}-4za^{15}+a^{15}z^{-1}+2z^{8}a^{14}-10z^{6}a^{14}+13z^{4}a^{14}-7z^{2}a^{14}-a^{14}z^{-2}+3a^{14}+z^{9}a^{13}-2z^{7}a^{13}-11z^{5}a^{13}+26z^{3}a^{13}-19za^{13}+5a^{13}z^{-1}+4z^{8}a^{12}-23z^{6}a^{12}+38z^{4}a^{12}-29z^{2}a^{12}-4a^{12}z^{-2}+15a^{12}+z^{9}a^{11}-2z^{7}a^{11}-16z^{5}a^{11}+42z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+3z^{8}a^{10}-21z^{6}a^{10}+44z^{4}a^{10}-42z^{2}a^{10}-5a^{10}z^{-2}+20a^{10}+2z^{7}a^{9}-15z^{5}a^{9}+27z^{3}a^{9}-18za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-7z^{6}a^{8}+14z^{4}a^{8}-15z^{2}a^{8}-2a^{8}z^{-2}+8a^{8}+z^{7}a^{7}-5z^{5}a^{7}+4z^{3}a^{7}+z^{6}a^{6}-5z^{4}a^{6}+5z^{2}a^{6}-a^{6}$ (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

-5

1

-7

1

0

-9

1

0

-11

2

1

2

-13

2

4

1

-15

3

1

3

-17

2

5

2

1

-19

2

1

-21

1

2

1

0

-23

1

2

-1

-25

1

-27

1

-1

Integral Khovanov Homology

(db, data source)

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