L10a50

L10a49

L10a51

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a50 at Knotilus!

Link Presentations

[edit Notes on L10a50's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {6uv^{2}-7uv+2u+2v^{3}-7v^{2}+6v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {3}{q^{21/2}}}-{\frac {1}{q^{23/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{11}z-a^{11}z^{-1}-a^{9}z^{3}+2a^{9}z+2a^{9}z^{-1}-3a^{7}z^{3}-2a^{7}z-3a^{5}z^{3}-3a^{5}z-a^{5}z^{-1}-a^{3}z^{3}$ (db)
Kauffman polynomial	$-z^{7}a^{13}+4z^{5}a^{13}-4z^{3}a^{13}+za^{13}-3z^{8}a^{12}+14z^{6}a^{12}-19z^{4}a^{12}+6z^{2}a^{12}+2a^{12}-2z^{9}a^{11}+5z^{7}a^{11}+3z^{5}a^{11}-9z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-8z^{8}a^{10}+30z^{6}a^{10}-29z^{4}a^{10}+2z^{2}a^{10}+5a^{10}-2z^{9}a^{9}-z^{7}a^{9}+17z^{5}a^{9}-15z^{3}a^{9}+4za^{9}-2a^{9}z^{-1}-5z^{8}a^{8}+9z^{6}a^{8}+z^{4}a^{8}-4z^{2}a^{8}+3a^{8}-7z^{7}a^{7}+12z^{5}a^{7}-3z^{3}a^{7}-7z^{6}a^{6}+8z^{4}a^{6}-a^{6}-6z^{5}a^{5}+6z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-3z^{4}a^{4}-z^{3}a^{3}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                     1 1 -4                   3 1 -2 -6                 3     3 -8               4 3     -1 -10             6 3       3 -12           4 5         1 -14         5 5           0 -16       2 4             2 -18     2 5               -3 -20   1 2                 1 -22   2                   -2 -24 1                     1

Integral Khovanov Homology (db, data source)

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

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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