L10a29

L10a28

L10a30

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a29 at Knotilus!

Link Presentations

[edit Notes on L10a29's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,8,15,7 X20,12,5,11 X8,20,9,19 X16,9,17,10 X18,15,19,16 X10,17,11,18 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -5, 6, -8, 4, -2, 10, -3, 7, -6, 8, -7, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-v+1\right)^{2}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+6q^{7/2}-9q^{5/2}+11q^{3/2}-13{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-2a^{3}z+3za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+7az^{3}-10z^{3}a^{-1}+8az-9za^{-1}+4az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+3z^{5}a^{-5}-3z^{3}a^{-5}+a^{4}z^{6}+5z^{6}a^{-4}-3a^{4}z^{4}-6z^{4}a^{-4}+3a^{4}z^{2}+3z^{2}a^{-4}-a^{4}-a^{-4}+3a^{3}z^{7}+6z^{7}a^{-3}-9a^{3}z^{5}-10z^{5}a^{-3}+8a^{3}z^{3}+11z^{3}a^{-3}-4a^{3}z-6za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+3a^{2}z^{8}+4z^{8}a^{-2}-4a^{2}z^{6}-z^{6}a^{-2}-8a^{2}z^{4}-9z^{4}a^{-2}+11a^{2}z^{2}+12z^{2}a^{-2}-4a^{2}-4a^{-2}+az^{9}+z^{9}a^{-1}+8az^{7}+11z^{7}a^{-1}-31az^{5}-35z^{5}a^{-1}+32az^{3}+38z^{3}a^{-1}-15az-17za^{-1}+4az^{-1}+4a^{-1}z^{-1}+7z^{8}-11z^{6}-7z^{4}+16z^{2}-7$ (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                 4 1   -3 6               5 2     3 4             6 4       -2 2           7 5         2 0         5 8           3 -2       4 5             -1 -4     2 5               3 -6   1 4                 -3 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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.

L10a28 L10a30