L10a106

L10a105

L10a107

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a106 at Knotilus!

Link Presentations

[edit Notes on L10a106's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}+t(2)t(1)^{3}-3t(2)^{3}t(1)^{2}+7t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-5t(2)^{2}t(1)+7t(2)t(1)-3t(1)+t(2)^{2}-2t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {13}{q^{9/2}}}-{\frac {14}{q^{7/2}}}+{\frac {13}{q^{5/2}}}+q^{3/2}-{\frac {11}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {11}{q^{11/2}}}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z+2a^{5}z^{5}+6a^{5}z^{3}+5a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-6a^{3}z^{3}-5a^{3}z-a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-za^{9}-5z^{6}a^{8}+4z^{4}a^{8}-z^{2}a^{8}-7z^{7}a^{7}+10z^{5}a^{7}-8z^{3}a^{7}+za^{7}-6z^{8}a^{6}+7z^{6}a^{6}-2z^{4}a^{6}-z^{2}a^{6}-2z^{9}a^{5}-10z^{7}a^{5}+33z^{5}a^{5}-30z^{3}a^{5}+10za^{5}-a^{5}z^{-1}-11z^{8}a^{4}+24z^{6}a^{4}-12z^{4}a^{4}+z^{2}a^{4}+a^{4}-2z^{9}a^{3}-7z^{7}a^{3}+31z^{5}a^{3}-27z^{3}a^{3}+9za^{3}-a^{3}z^{-1}-5z^{8}a^{2}+11z^{6}a^{2}-3z^{4}a^{2}-z^{2}a^{2}-4z^{7}a+11z^{5}a-8z^{3}a+za-z^{6}+2z^{4}-z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   3   3 0                 4 1   -3 -2               7 3     4 -4             7 5       -2 -6           7 6         1 -8         6 7           1 -10       5 7             -2 -12     2 7               5 -14   1 4                 -3 -16   2                   2 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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.

L10a105 L10a107