L11n239

L11n238

L11n240

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n239 at Knotilus!

Link Presentations

[edit Notes on L11n239's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,16,8,17 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X2,11,3,12 X17,8,18,9
Gauss code	{1, -10, -4, 5, -3, 4, -2, 11, -5, 3}, {10, -1, 6, -9, 8, 2, -11, -6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2u^{3}v+u^{3}+2u^{2}v^{2}-2u^{2}v-2uv^{2}+2uv+v^{3}-2v^{2}}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+2q^{5/2}-2q^{3/2}+{\sqrt {q}}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$za^{5}+2a^{5}z^{-1}-2z^{3}a^{3}-7za^{3}-5a^{3}z^{-1}+z^{5}a+6z^{3}a+10za+6az^{-1}-z^{5}a^{-1}-5z^{3}a^{-1}-8za^{-1}-4a^{-1}z^{-1}+z^{3}a^{-3}+2za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-z^{8}a^{-2}-z^{8}-a^{5}z^{7}-3a^{3}z^{7}-3az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}+5a^{4}z^{6}+6a^{2}z^{6}+3z^{6}a^{-2}-z^{6}a^{-4}+5z^{6}+6a^{5}z^{5}+19a^{3}z^{5}+22az^{5}+18z^{5}a^{-1}+9z^{5}a^{-3}-5a^{4}z^{4}-6a^{2}z^{4}+4z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}-11a^{5}z^{3}-33a^{3}z^{3}-42az^{3}-29z^{3}a^{-1}-9z^{3}a^{-3}-a^{2}z^{2}-9z^{2}a^{-2}-3z^{2}a^{-4}-7z^{2}+8a^{5}z+22a^{3}z+28az+18za^{-1}+4za^{-3}+a^{4}+a^{2}+3a^{-2}+a^{-4}+3-2a^{5}z^{-1}-5a^{3}z^{-1}-6az^{-1}-4a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     1   1 6                   1 1   0 4               1 2 1     0 2             1 1 1       1 0           1 3 2         0 -2         1 2 2           1 -4       1 1 1             -1 -6     1 1 1               1 -8   1 2                   1 -10                         0 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11n238 L11n240