L11n32

L11n31

L11n33

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n32 at Knotilus!

Link Presentations

[edit Notes on L11n32's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X3849 X5,13,6,12 X13,5,14,22 X15,18,16,19 X11,17,12,16 X17,11,18,10 X19,2,20,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, -11, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+5t(1)t(2)^{3}-t(2)^{3}-t(1)t(2)^{2}+5t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{5/2}+2q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {5}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+a^{5}z^{5}+4a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-9a^{3}z^{3}-10a^{3}z-4a^{3}z^{-1}+2az^{5}+8az^{3}-z^{3}a^{-1}+8az+3az^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-4a^{4}z^{8}-6a^{2}z^{8}-2z^{8}-5a^{5}z^{7}-5a^{3}z^{7}-az^{7}-z^{7}a^{-1}-2a^{6}z^{6}+13a^{4}z^{6}+24a^{2}z^{6}+9z^{6}+18a^{5}z^{5}+32a^{3}z^{5}+19az^{5}+5z^{5}a^{-1}+2a^{6}z^{4}-13a^{4}z^{4}-27a^{2}z^{4}-12z^{4}-4a^{7}z^{3}-25a^{5}z^{3}-44a^{3}z^{3}-31az^{3}-8z^{3}a^{-1}-a^{8}z^{2}-a^{6}z^{2}+7a^{4}z^{2}+13a^{2}z^{2}+6z^{2}+2a^{7}z+14a^{5}z+24a^{3}z+17az+5za^{-1}-a^{6}-2a^{4}-3a^{2}-1-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                 1   -1 2               3 1   2 0             3 1     -2 -2           4 3       1 -4         4 4         0 -6       3 3           0 -8     2 4             2 -10   2 3               -1 -12   3                 3 -14 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n31 L11n33