L10a43

L10a42

L10a44

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a43 at Knotilus!

Link Presentations

[edit Notes on L10a43's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{5}-4uv^{4}+8uv^{3}-8uv^{2}+4uv+4v^{4}-8v^{3}+8v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-9q^{9/2}+13q^{7/2}-{\frac {1}{q^{7/2}}}-16q^{5/2}+{\frac {3}{q^{5/2}}}+17q^{3/2}-{\frac {8}{q^{3/2}}}-q^{13/2}+5q^{11/2}-16{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-5z^{5}a^{-1}+2z^{5}a^{-3}+3az^{3}-12z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+5az-12za^{-1}+5za^{-3}+3az^{-1}-5a^{-1}z^{-1}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}+5z^{6}a^{-6}-6z^{4}a^{-6}+a^{-6}+9z^{7}a^{-5}-15z^{5}a^{-5}+5z^{3}a^{-5}+7z^{8}a^{-4}-4z^{6}a^{-4}-9z^{4}a^{-4}+4z^{2}a^{-4}+2z^{9}a^{-3}+15z^{7}a^{-3}+a^{3}z^{5}-34z^{5}a^{-3}-2a^{3}z^{3}+18z^{3}a^{-3}+a^{3}z-5za^{-3}+2a^{-3}z^{-1}+12z^{8}a^{-2}+3a^{2}z^{6}-13z^{6}a^{-2}-4a^{2}z^{4}-10z^{4}a^{-2}+a^{2}z^{2}+17z^{2}a^{-2}-5a^{-2}+2z^{9}a^{-1}+6az^{7}+12z^{7}a^{-1}-12az^{5}-31z^{5}a^{-1}+14az^{3}+29z^{3}a^{-1}-10az-16za^{-1}+3az^{-1}+5a^{-1}z^{-1}+5z^{8}-z^{6}-11z^{4}+14z^{2}-5$ (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 14                     1 1 12                   4   -4 10                 5 1   4 8               8 4     -4 6             8 5       3 4           9 8         -1 2         7 8           -1 0       5 10             5 -2     3 6               -3 -4     5                 5 -6 1 3                   -2 -8 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

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