L11a320

L11a319

L11a321

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a320 at Knotilus!

Link Presentations

[edit Notes on L11a320's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X14,8,15,7 X16,14,17,13 X8,16,1,15 X22,17,9,18 X18,21,19,22 X6,9,7,10 X4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, 5, -7}, {10, -1, 2, -3, 6, -5, 7, -6, 8, -9, 11, -4, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)(2t(2)t(1)-t(1)-t(2)+2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {14}{q^{5/2}}}-{\frac {18}{q^{7/2}}}+{\frac {17}{q^{9/2}}}-{\frac {15}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+3a^{7}z^{3}+3a^{7}z-a^{5}z^{7}-4a^{5}z^{5}-6a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-6a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}+az^{5}+3az^{3}+2az$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+7z^{5}a^{9}-2z^{3}a^{9}-6z^{8}a^{8}+10z^{6}a^{8}-10z^{4}a^{8}+7z^{2}a^{8}-4z^{9}a^{7}+2z^{7}a^{7}+z^{5}a^{7}+z^{3}a^{7}-2za^{7}-z^{10}a^{6}-10z^{8}a^{6}+27z^{6}a^{6}-30z^{4}a^{6}+13z^{2}a^{6}-7z^{9}a^{5}+11z^{7}a^{5}-5z^{5}a^{5}+z^{3}a^{5}-za^{5}-a^{5}z^{-1}-z^{10}a^{4}-8z^{8}a^{4}+24z^{6}a^{4}-22z^{4}a^{4}+6z^{2}a^{4}+a^{4}-3z^{9}a^{3}+z^{7}a^{3}+11z^{5}a^{3}-12z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-4z^{8}a^{2}+9z^{6}a^{2}-4z^{4}a^{2}-3z^{7}a+9z^{5}a-8z^{3}a+3za-z^{6}+3z^{4}-2z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   4 1   -3 -2                 7 2     5 -4               8 5       -3 -6             10 6         4 -8           7 8           1 -10         8 10             -2 -12       5 8               3 -14     2 7                 -5 -16   1 5                   4 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a319 L11a321