L10a3

L10a2

L10a4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a3 at Knotilus!

Link Presentations

[edit Notes on L10a3's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X14,10,15,9 X8493 X10,5,11,6 X20,11,5,12 X18,13,19,14 X12,19,13,20 X2,16,3,15
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -9, 8, -4, 10, -2, 3, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-2)(t(2)-1)(2t(2)-1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+6{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+2za^{5}-z^{5}a^{3}-z^{3}a^{3}-za^{3}-z^{5}a-z^{3}a+az^{-1}+z^{3}a^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{5}z^{9}-2a^{3}z^{9}-4a^{6}z^{8}-9a^{4}z^{8}-5a^{2}z^{8}-3a^{7}z^{7}-3a^{3}z^{7}-6az^{7}-a^{8}z^{6}+13a^{6}z^{6}+27a^{4}z^{6}+7a^{2}z^{6}-6z^{6}+10a^{7}z^{5}+14a^{5}z^{5}+13a^{3}z^{5}+5az^{5}-4z^{5}a^{-1}+3a^{8}z^{4}-12a^{6}z^{4}-28a^{4}z^{4}-6a^{2}z^{4}-z^{4}a^{-2}+6z^{4}-8a^{7}z^{3}-17a^{5}z^{3}-11a^{3}z^{3}+2az^{3}+4z^{3}a^{-1}-a^{8}z^{2}+4a^{6}z^{2}+9a^{4}z^{2}+4a^{2}z^{2}+2a^{7}z+4a^{5}z+2a^{3}z+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                     1 -1 4                   3   3 2                 3 1   -2 0               7 3     4 -2             6 5       -1 -4           6 5         1 -6         5 6           1 -8       3 6             -3 -10     2 5               3 -12   1 3                 -2 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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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