L11a11

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a11 at Knotilus!

Link Presentations

[edit Notes on L11a11's Link Presentations]

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

A Braid Representative
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	$-{\frac {16}{q^{9/2}}}-q^{7/2}+{\frac {23}{q^{7/2}}}+5q^{5/2}-{\frac {27}{q^{5/2}}}-12q^{3/2}+{\frac {27}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+18{\sqrt {q}}-{\frac {25}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+3a^{5}z-3a^{3}z^{5}-6a^{3}z^{3}-4a^{3}z+az^{7}+3az^{5}-z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}+3az+az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+7a^{6}z^{8}-14a^{6}z^{6}+10a^{6}z^{4}-3a^{6}z^{2}+6a^{5}z^{9}-22a^{5}z^{5}+23a^{5}z^{3}-6a^{5}z+2a^{4}z^{10}+20a^{4}z^{8}-53a^{4}z^{6}+41a^{4}z^{4}-10a^{4}z^{2}+15a^{3}z^{9}-9a^{3}z^{7}-34a^{3}z^{5}+z^{5}a^{-3}+35a^{3}z^{3}-8a^{3}z+2a^{2}z^{10}+28a^{2}z^{8}-64a^{2}z^{6}+5z^{6}a^{-2}+41a^{2}z^{4}-3z^{4}a^{-2}-9a^{2}z^{2}+9az^{9}+7az^{7}+12z^{7}a^{-1}-38az^{5}-16z^{5}a^{-1}+26az^{3}+7z^{3}a^{-1}-6az-2za^{-1}+az^{-1}+a^{-1}z^{-1}+15z^{8}-21z^{6}+9z^{4}-3z^{2}-1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   8 1   7 2                 10 4     -6 0               15 8       7 -2             14 12         -2 -4           13 13           0 -6         10 14             4 -8       6 13               -7 -10     3 10                 7 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

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