L10a122

L10a121

L10a123

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a122 at Knotilus!

Link Presentations

[edit Notes on L10a122's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+4t(1)t(3)^{2}-3t(1)t(2)t(3)^{2}+6t(2)t(3)^{2}-3t(3)^{2}-6t(1)t(3)+3t(1)t(2)t(3)-4t(2)t(3)+3t(3)+2t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-2q^{-5}+q^{4}+6q^{-4}-4q^{3}-9q^{-3}+9q^{2}+13q^{-2}-11q-14q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-3a^{4}z^{-2}-5a^{4}+3z^{4}a^{2}+7z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-z^{6}-3z^{4}-6z^{2}-3z^{-2}-6+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+3a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+2a^{4}z^{8}+a^{4}z^{6}-16a^{4}z^{4}+z^{4}a^{-4}+24a^{4}z^{2}+3a^{4}z^{-2}-14a^{4}+a^{3}z^{9}+7a^{3}z^{7}-20a^{3}z^{5}+4z^{5}a^{-3}+12a^{3}z^{3}-z^{3}a^{-3}+a^{3}z-a^{3}z^{-1}+7a^{2}z^{8}-2a^{2}z^{6}+9z^{6}a^{-2}-30a^{2}z^{4}-11z^{4}a^{-2}+40a^{2}z^{2}+6z^{2}a^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}-21a^{2}-4a^{-2}+az^{9}+15az^{7}+10z^{7}a^{-1}-32az^{5}-13z^{5}a^{-1}+14az^{3}+4z^{3}a^{-1}+az+za^{-1}-az^{-1}-a^{-1}z^{-1}+5z^{8}+7z^{6}-30z^{4}+28z^{2}+3z^{-2}-14$ (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 9                     1 1 7                   3   -3 5                 6 1   5 3               5 3     -2 1             9 6       3 -1           8 8         0 -3         5 6           -1 -5       4 8             4 -7     2 5               -3 -9   1 5                 4 -11   1                   -1 -13 1                     1

Integral Khovanov Homology (db, data source)

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

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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