L11n21

L11n20

L11n22

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n21 at Knotilus!

Link Presentations

[edit Notes on L11n21's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X13,22,14,5 X21,14,22,15 X9,18,10,19 X11,20,12,21 X19,10,20,11 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -8, 10, -9, 4, -6, 7, 11, -2, 3, 8, -10, 9, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {4}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {10}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-2a^{9}z-2a^{9}z^{-1}+a^{7}z^{5}+3a^{7}z^{3}+6a^{7}z+4a^{7}z^{-1}+a^{5}z^{5}+a^{5}z^{3}-a^{5}z-a^{5}z^{-1}-2a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+5a^{12}z^{2}-2a^{12}+2a^{11}z^{7}-6a^{11}z^{5}+4a^{11}z^{3}+a^{11}z+2a^{10}z^{8}-3a^{10}z^{6}-4a^{10}z^{4}+6a^{10}z^{2}-a^{10}+a^{9}z^{9}+a^{9}z^{7}-3a^{9}z^{5}-6a^{9}z^{3}+6a^{9}z-2a^{9}z^{-1}+4a^{8}z^{8}-6a^{8}z^{6}+4a^{8}z^{4}-11a^{8}z^{2}+6a^{8}+a^{7}z^{9}+a^{7}z^{7}+4a^{7}z^{5}-15a^{7}z^{3}+11a^{7}z-4a^{7}z^{-1}+2a^{6}z^{8}-a^{6}z^{6}+6a^{6}z^{4}-11a^{6}z^{2}+5a^{6}+2a^{5}z^{7}+a^{5}z^{5}-2a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}+a^{4}z^{6}+2a^{4}z^{4}+a^{4}z^{2}-a^{4}+3a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 4 2 -2 -6               4     4 -8             4 4     0 -10           6 4       2 -12         3 4         1 -14       4 6           -2 -16     1 3             2 -18   1 4               -3 -20   1                 1 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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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