L10a42

L10a41

L10a43

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a42 at Knotilus!

Link Presentations

[edit Notes on L10a42's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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