L10a109

L10a108

L10a110

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a109 at Knotilus!

Link Presentations

[edit Notes on L10a109's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,14,19,13 X20,16,11,15 X14,20,15,19 X2,11,3,12 X8,18,9,17
Gauss code	{1, -9, 4, -5, 3, -4, 2, -10, 5, -3}, {9, -1, 6, -8, 7, -2, 10, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}+u^{2}-2uv^{2}+uv-2u+v^{2}+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+7q^{7/2}-10q^{5/2}+11q^{3/2}-13{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-3a^{3}z+4za^{-3}-2a^{3}z^{-1}+2a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+8az^{3}-11z^{3}a^{-1}+12az-13za^{-1}+7az^{-1}-7a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+3z^{5}a^{-5}-2z^{3}a^{-5}+a^{4}z^{6}+6z^{6}a^{-4}-4a^{4}z^{4}-8z^{4}a^{-4}+5a^{4}z^{2}+6z^{2}a^{-4}-2a^{4}-2a^{-4}+2a^{3}z^{7}+7z^{7}a^{-3}-6a^{3}z^{5}-11z^{5}a^{-3}+6a^{3}z^{3}+10z^{3}a^{-3}-4a^{3}z-6za^{-3}+2a^{3}z^{-1}+2a^{-3}z^{-1}+2a^{2}z^{8}+4z^{8}a^{-2}-a^{2}z^{6}+2z^{6}a^{-2}-11a^{2}z^{4}-17z^{4}a^{-2}+17a^{2}z^{2}+20z^{2}a^{-2}-8a^{2}-8a^{-2}+az^{9}+z^{9}a^{-1}+5az^{7}+10z^{7}a^{-1}-19az^{5}-27z^{5}a^{-1}+21az^{3}+27z^{3}a^{-1}-16az-18za^{-1}+7az^{-1}+7a^{-1}z^{-1}+6z^{8}-6z^{6}-15z^{4}+25z^{2}-13$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   2   2 8                 5 1   -4 6               5 2     3 4             6 5       -1 2           7 5         2 0         5 8           3 -2       4 5             -1 -4     1 5               4 -6   1 4                 -3 -8   1                   1 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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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