L11n39

L11n38

L11n40

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n39 at Knotilus!

Link Presentations

[edit Notes on L11n39's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X5,12,6,13 X8493 X9,16,10,17 X13,19,14,18 X17,15,18,14 X15,10,16,11 X11,22,12,5 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 9, -10, 4, -7, 8, -9, 6, -8, 7, 11, -2, 3, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-q^{5/2}-{\frac {2}{q^{5/2}}}+2q^{3/2}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-2{\sqrt {q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+2a^{5}z^{3}+5a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-8a^{3}z-3a^{3}z^{-1}+az^{5}+5az^{3}-z^{3}a^{-1}+6az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+4a^{8}z^{2}-a^{8}+2a^{7}z^{7}-8a^{7}z^{5}+7a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-2a^{6}z^{6}-5a^{6}z^{4}+5a^{6}z^{2}-2a^{6}+3a^{5}z^{7}-15a^{5}z^{5}+19a^{5}z^{3}-12a^{5}z+3a^{5}z^{-1}+2a^{4}z^{8}-12a^{4}z^{6}+17a^{4}z^{4}-7a^{4}z^{2}+a^{3}z^{9}-4a^{3}z^{7}-7a^{3}z^{5}+27a^{3}z^{3}-18a^{3}z+3a^{3}z^{-1}+3a^{2}z^{8}-20a^{2}z^{6}+32a^{2}z^{4}-13a^{2}z^{2}+2a^{2}+az^{9}-4az^{7}+z^{7}a^{-1}-5az^{5}-5z^{5}a^{-1}+21az^{3}+6z^{3}a^{-1}-13az-4za^{-1}+2az^{-1}+a^{-1}z^{-1}+2z^{8}-11z^{6}+14z^{4}-5z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     1   -1 2                   1 1   0 0               3 2 1     2 -2             2 3 1       0 -4           2 2 2         2 -6         2 3 1           0 -8       2 2 1             1 -10     1 2 1               0 -12   1 2                   -1 -14   1                     1 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11n38 L11n40