L10a104

L10a103

L10a105

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a104 at Knotilus!

Link Presentations

[edit Notes on L10a104's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,11,17,12 X8,9,1,10 X20,17,9,18 X12,4,13,3 X14,8,15,7 X6,14,7,13 X18,6,19,5 X4,20,5,19 X2,16,3,15
Gauss code	{1, -10, 5, -9, 8, -7, 6, -3}, {3, -1, 2, -5, 7, -6, 10, -2, 4, -8, 9, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}\left(-v^{3}\right)+2u^{3}v^{2}-u^{3}v+3u^{2}v^{3}-7u^{2}v^{2}+7u^{2}v-2u^{2}-2uv^{3}+7uv^{2}-7uv+3u-v^{2}+2v-1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-7q^{9/2}+11q^{7/2}-{\frac {1}{q^{7/2}}}-15q^{5/2}+{\frac {4}{q^{5/2}}}+15q^{3/2}-{\frac {8}{q^{3/2}}}-q^{13/2}+3q^{11/2}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}+6z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2az^{3}-6z^{3}a^{-1}+az-3za^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+za^{-7}+3z^{6}a^{-6}-5z^{4}a^{-6}+2z^{2}a^{-6}+5z^{7}a^{-5}-8z^{5}a^{-5}+6z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-5z^{6}a^{-4}+2z^{2}a^{-4}-a^{-4}+2z^{9}a^{-3}+9z^{7}a^{-3}+a^{3}z^{5}-25z^{5}a^{-3}-a^{3}z^{3}+23z^{3}a^{-3}-9za^{-3}+a^{-3}z^{-1}+11z^{8}a^{-2}+4a^{2}z^{6}-17z^{6}a^{-2}-6a^{2}z^{4}+6z^{4}a^{-2}+2a^{2}z^{2}+z^{2}a^{-2}+2z^{9}a^{-1}+7az^{7}+11z^{7}a^{-1}-12az^{5}-29z^{5}a^{-1}+6az^{3}+22z^{3}a^{-1}-2az-6za^{-1}+6z^{8}-5z^{6}-5z^{4}+3z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   2   -2 10                 5 1   4 8               7 3     -4 6             8 4       4 4           7 7         0 2         8 8           0 0       5 8             3 -2     3 7               -4 -4   1 5                 4 -6   3                   -3 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L10a103 L10a105