L11n221

L11n220

L11n222

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n221 at Knotilus!

Link Presentations

[edit Notes on L11n221's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X12,4,13,3 X22,16,9,15 X2,17,3,18 X21,4,22,5 X5,15,6,14 X13,21,14,20 X16,12,17,11 X6,19,7,20 X18,7,19,8
Gauss code	{1, -5, 3, 6, -7, -10, 11, -2}, {2, -1, 9, -3, -8, 7, 4, -9, 5, -11, 10, 8, -6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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