L10a113

L10a112

L10a114

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a113 at Knotilus!

Link Presentations

[edit Notes on L10a113's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X18,14,19,13 X20,8,11,7 X6,20,7,19 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17
Gauss code	{1, -9, 2, -6, 7, -5, 4, -2, 8, -10}, {9, -1, 3, -7, 6, -8, 10, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)\left(t(1)^{2}-t(1)+1\right)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$10q^{9/2}-11q^{7/2}+11q^{5/2}-{\frac {1}{q^{5/2}}}-12q^{3/2}+{\frac {3}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+8{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-7z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+2az-7za^{-1}+8za^{-3}-3za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-1}-2z^{9}a^{-3}-10z^{8}a^{-2}-7z^{8}a^{-4}-3z^{8}-az^{7}+z^{7}a^{-1}-9z^{7}a^{-3}-11z^{7}a^{-5}+32z^{6}a^{-2}+10z^{6}a^{-4}-10z^{6}a^{-6}+12z^{6}+4az^{5}+18z^{5}a^{-1}+44z^{5}a^{-3}+24z^{5}a^{-5}-6z^{5}a^{-7}-23z^{4}a^{-2}+10z^{4}a^{-4}+16z^{4}a^{-6}-3z^{4}a^{-8}-14z^{4}-6az^{3}-30z^{3}a^{-1}-44z^{3}a^{-3}-16z^{3}a^{-5}+3z^{3}a^{-7}-z^{3}a^{-9}-11z^{2}a^{-4}-8z^{2}a^{-6}+3z^{2}+4az+14za^{-1}+16za^{-3}+6za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 4 1   3 10               6 2     -4 8             5 4       1 6           6 6         0 4         6 5           1 2       4 8             4 0     2 4               -2 -2   1 4                 3 -4   2                   -2 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L10a112 L10a114