L11a333

L11a332

L11a334

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a333 at Knotilus!

Link Presentations

[edit Notes on L11a333's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,5,21,6 X16,7,17,8 X8,9,1,10 X18,12,19,11 X6,15,7,16 X4,14,5,13 X22,18,9,17 X2,19,3,20 X14,22,15,21
Gauss code	{1, -10, 2, -8, 3, -7, 4, -5}, {5, -1, 6, -2, 8, -11, 7, -4, 9, -6, 10, -3, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)t(2)+1)\left(t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+5t(1)t(2)-3t(2)-t(1)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+9q^{5/2}-14q^{3/2}+19{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {20}{q^{5/2}}}+{\frac {14}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+4a^{3}z^{5}+5a^{3}z^{3}+3a^{3}z+2a^{3}z^{-1}-az^{9}-6az^{7}+z^{7}a^{-1}-13az^{5}+4z^{5}a^{-1}-13az^{3}+5z^{3}a^{-1}-8az+3za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{10}-3z^{10}-8a^{3}z^{9}-15az^{9}-7z^{9}a^{-1}-10a^{4}z^{8}-9a^{2}z^{8}-7z^{8}a^{-2}-6z^{8}-8a^{5}z^{7}+11a^{3}z^{7}+37az^{7}+14z^{7}a^{-1}-4z^{7}a^{-3}-4a^{6}z^{6}+18a^{4}z^{6}+32a^{2}z^{6}+16z^{6}a^{-2}-z^{6}a^{-4}+27z^{6}-a^{7}z^{5}+13a^{5}z^{5}-5a^{3}z^{5}-37az^{5}-9z^{5}a^{-1}+9z^{5}a^{-3}+5a^{6}z^{4}-13a^{4}z^{4}-34a^{2}z^{4}-9z^{4}a^{-2}+2z^{4}a^{-4}-27z^{4}+a^{7}z^{3}-6a^{5}z^{3}+a^{3}z^{3}+19az^{3}+7z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+13a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+13z^{2}+2a^{5}z-4a^{3}z-9az-3za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   6 1   -5 4                 8 3     5 2               11 6       -5 0             12 8         4 -2           11 12           1 -4         9 11             -2 -6       5 11               6 -8     4 9                 -5 -10   1 6                   5 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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