L10a40

L10a39

L10a41

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a40 at Knotilus!

Link Presentations

[edit Notes on L10a40's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,8,19,7 X20,13,5,14 X14,19,15,20 X16,10,17,9 X10,16,11,15 X8,18,9,17 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 6, -7, 10, -2, 4, -5, 7, -6, 8, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(2)^{3}+4t(1)t(2)^{2}-7t(2)^{2}-7t(1)t(2)+4t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}-q^{7/2}+{\frac {6}{q^{7/2}}}+2q^{5/2}-{\frac {8}{q^{5/2}}}-4q^{3/2}+{\frac {9}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+6{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-a^{5}z^{-1}+2a^{3}z^{3}-a^{3}z^{-1}-za^{-3}+3az^{3}+z^{3}a^{-1}+3az+2az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+2a^{6}z^{6}-4a^{6}z^{4}+a^{6}+2a^{5}z^{7}-a^{5}z^{5}-6a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}+2a^{4}z^{8}-3a^{4}z^{6}+3a^{4}z^{4}-5a^{4}z^{2}+2a^{4}+a^{3}z^{9}+a^{3}z^{7}-7a^{3}z^{5}+z^{5}a^{-3}+13a^{3}z^{3}-3z^{3}a^{-3}-8a^{3}z+za^{-3}+a^{3}z^{-1}+5a^{2}z^{8}-17a^{2}z^{6}+2z^{6}a^{-2}+26a^{2}z^{4}-5z^{4}a^{-2}-13a^{2}z^{2}+z^{2}a^{-2}+3a^{2}+az^{9}+2az^{7}+3z^{7}a^{-1}-15az^{5}-9z^{5}a^{-1}+28az^{3}+9z^{3}a^{-1}-14az-5za^{-1}+2az^{-1}+a^{-1}z^{-1}+3z^{8}-10z^{6}+14z^{4}-7z^{2}+1$ (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 χ 8                     1 1 6                   1   -1 4                 3 1   2 2               3 1     -2 0             5 3       2 -2           5 4         -1 -4         3 4           -1 -6       3 5             2 -8     2 3               -1 -10   1 4                 3 -12   1                   -1 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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.

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