L10a2

L10a1

L10a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a2 at Knotilus!

Link Presentations

[edit Notes on L10a2's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-3t(2)^{3}+3t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {14}{q^{9/2}}}-{\frac {15}{q^{7/2}}}+{\frac {13}{q^{5/2}}}+q^{3/2}-{\frac {11}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {11}{q^{11/2}}}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-a^{7}z+2a^{5}z^{5}+5a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-5a^{3}z^{3}+3a^{3}z^{-1}+az^{5}+2az^{3}-az-2az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}+4a^{9}z^{5}-2a^{9}z^{3}+8a^{8}z^{6}-9a^{8}z^{4}+3a^{8}z^{2}+9a^{7}z^{7}-11a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z+6a^{6}z^{8}-a^{6}z^{6}-9a^{6}z^{4}+2a^{6}z^{2}+a^{6}+2a^{5}z^{9}+11a^{5}z^{7}-30a^{5}z^{5}+20a^{5}z^{3}-4a^{5}z-a^{5}z^{-1}+11a^{4}z^{8}-23a^{4}z^{6}+12a^{4}z^{4}-4a^{4}z^{2}+3a^{4}+2a^{3}z^{9}+6a^{3}z^{7}-27a^{3}z^{5}+22a^{3}z^{3}-a^{3}z-3a^{3}z^{-1}+5a^{2}z^{8}-13a^{2}z^{6}+9a^{2}z^{4}-3a^{2}z^{2}+3a^{2}+4az^{7}-12az^{5}+9az^{3}+az-2az^{-1}+z^{6}-2z^{4}$ (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 4                     1 -1 2                   3   3 0                 3 1   -2 -2               8 3     5 -4             7 5       -2 -6           8 6         2 -8         6 7           1 -10       5 8             -3 -12     3 6               3 -14   1 5                 -4 -16   3                   3 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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