L10a147

L10a146

L10a148

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a147 at Knotilus!

Link Presentations

[edit Notes on L10a147's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X20,10,13,9 X18,8,19,7 X8,14,9,13 X12,15,5,16 X10,20,11,19 X2,11,3,12 X4,18,1,17
Gauss code	{1, -9, 2, -10}, {3, -1, 5, -6, 4, -8, 9, -7}, {6, -2, 7, -3, 10, -5, 8, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}-t(1)t(3)t(2)^{3}+t(3)t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-3t(1)t(3)^{2}t(2)^{2}+3t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+4t(1)t(3)t(2)^{2}-3t(3)t(2)^{2}+t(2)^{2}-t(1)t(3)^{3}t(2)+t(3)^{3}t(2)+3t(1)t(3)^{2}t(2)-4t(3)^{2}t(2)+t(1)t(2)-3t(1)t(3)t(2)+3t(3)t(2)-t(2)-t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}+6q^{4}+q^{-4}-9q^{3}-4q^{-3}+14q^{2}+8q^{-2}-13q-11q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-4z^{2}a^{-2}+2z^{2}a^{-4}-3a^{-2}+a^{-4}+2-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+10z^{8}a^{-2}+4z^{8}a^{-4}+6z^{8}+9az^{7}+8z^{7}a^{-1}+2z^{7}a^{-3}+3z^{7}a^{-5}+8a^{2}z^{6}-23z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-3z^{6}+4a^{3}z^{5}-11az^{5}-19z^{5}a^{-1}-13z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-9a^{2}z^{4}+20z^{4}a^{-2}+11z^{4}a^{-4}-3z^{4}a^{-6}-4z^{4}-2a^{3}z^{3}+3az^{3}+8z^{3}a^{-1}+10z^{3}a^{-3}+7z^{3}a^{-5}+3a^{2}z^{2}-13z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}-3za^{-1}-3za^{-3}+5a^{-2}+3a^{-4}+3+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                     1 1 11                   2   -2 9                 4 1   3 7               6 3     -3 5             8 3       5 3           5 6         1 1         9 8           1 -1       5 8             3 -3     3 6               -3 -5   1 5                 4 -7   3                   -3 -9 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L10a146 L10a148