L11a2

L11a1

L11a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a2 at Knotilus!

Link Presentations

[edit Notes on L11a2's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X14,10,15,9 X8493 X10,5,11,6 X18,11,19,12 X20,14,21,13 X22,19,5,20 X12,22,13,21 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 8, -4, 11, -2, 3, -7, 9, -8, 10, -9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-5t(2)^{3}+9t(2)^{2}-5t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {11}{q^{9/2}}}-5q^{7/2}+{\frac {17}{q^{7/2}}}+10q^{5/2}-{\frac {24}{q^{5/2}}}-17q^{3/2}+{\frac {27}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+23{\sqrt {q}}-{\frac {27}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-3a^{3}z^{3}+5az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-a^{3}z+az+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-8a^{3}z^{9}-15az^{9}-7z^{9}a^{-1}-13a^{4}z^{8}-26a^{2}z^{8}-9z^{8}a^{-2}-22z^{8}-11a^{5}z^{7}-8a^{3}z^{7}+9az^{7}+z^{7}a^{-1}-5z^{7}a^{-3}-5a^{6}z^{6}+16a^{4}z^{6}+58a^{2}z^{6}+19z^{6}a^{-2}-z^{6}a^{-4}+57z^{6}-a^{7}z^{5}+15a^{5}z^{5}+35a^{3}z^{5}+32az^{5}+23z^{5}a^{-1}+10z^{5}a^{-3}+4a^{6}z^{4}-5a^{4}z^{4}-35a^{2}z^{4}-12z^{4}a^{-2}+z^{4}a^{-4}-39z^{4}-6a^{5}z^{3}-21a^{3}z^{3}-27az^{3}-17z^{3}a^{-1}-5z^{3}a^{-3}+a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}+7z^{2}+a^{3}z-2za^{-1}-za^{-3}+1+a^{3}z^{-1}+2az^{-1}+2a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     4   4 6                   6 1   -5 4                 11 4     7 2               12 6       -6 0             15 11         4 -2           14 14           0 -4         10 13             -3 -6       7 14               7 -8     4 10                 -6 -10   1 7                   6 -12   4                     -4 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a1 L11a3