L11n45

L11n44

L11n46

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n45 at Knotilus!

Link Presentations

[edit Notes on L11n45's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X5,14,6,15 X3849 X9,18,10,19 X11,20,12,21 X13,22,14,5 X19,10,20,11 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, 11, -2, -3, 6, -9, 7, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)\left(t(2)^{2}+t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-5z^{3}a^{9}-7za^{9}-3a^{9}z^{-1}+z^{7}a^{7}+7z^{5}a^{7}+18z^{3}a^{7}+20za^{7}+7a^{7}z^{-1}-2z^{5}a^{5}-10z^{3}a^{5}-13za^{5}-4a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-3a^{14}z^{2}+a^{14}+a^{13}z^{5}-2a^{13}z^{3}+a^{12}z^{6}-2a^{12}z^{4}+a^{12}z^{2}+a^{11}z^{7}-3a^{11}z^{5}+4a^{11}z^{3}+a^{10}z^{8}-4a^{10}z^{6}+6a^{10}z^{4}+a^{9}z^{9}-6a^{9}z^{7}+15a^{9}z^{5}-16a^{9}z^{3}+10a^{9}z-3a^{9}z^{-1}+2a^{8}z^{8}-11a^{8}z^{6}+24a^{8}z^{4}-23a^{8}z^{2}+7a^{8}+a^{7}z^{9}-7a^{7}z^{7}+22a^{7}z^{5}-36a^{7}z^{3}+26a^{7}z-7a^{7}z^{-1}+a^{6}z^{8}-6a^{6}z^{6}+15a^{6}z^{4}-19a^{6}z^{2}+7a^{6}+3a^{5}z^{5}-14a^{5}z^{3}+16a^{5}z-4a^{5}z^{-1}$ (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 -4                   2 2 -6                 1 2 1 -8               2     2 -10             1 1     0 -12           3 2       1 -14         1 1         0 -16       2 3           -1 -18       1             1 -20   1 2               -1 -22                     0 -24 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n44 L11n46