L10a1

L9n28

L10a2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a1 at Knotilus!

Link Presentations

[edit Notes on L10a1's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-3v^{3}+5v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}-q^{7/2}+{\frac {13}{q^{7/2}}}+4q^{5/2}-{\frac {17}{q^{5/2}}}-8q^{3/2}+{\frac {17}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+12{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-z^{5}a^{-1}+a^{5}z^{3}-4a^{3}z^{3}+6az^{3}-2z^{3}a^{-1}+az-za^{-1}-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-2a^{3}z^{9}-2az^{9}-7a^{4}z^{8}-13a^{2}z^{8}-6z^{8}-9a^{5}z^{7}-17a^{3}z^{7}-15az^{7}-7z^{7}a^{-1}-5a^{6}z^{6}+3a^{4}z^{6}+15a^{2}z^{6}-4z^{6}a^{-2}+3z^{6}-a^{7}z^{5}+15a^{5}z^{5}+40a^{3}z^{5}+37az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+6a^{6}z^{4}+10a^{4}z^{4}+5a^{2}z^{4}+6z^{4}a^{-2}+7z^{4}-6a^{5}z^{3}-23a^{3}z^{3}-26az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-2a^{4}z^{2}-5a^{2}z^{2}-2z^{2}a^{-2}-5z^{2}+a^{3}z+3az+2za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   3   -3 4                 5 1   4 2               7 3     -4 0             10 5       5 -2           9 9         0 -4         8 8           0 -6       5 9             4 -8     4 8               -4 -10   1 5                 4 -12   4                   -4 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L9n28 L10a2