L11a16

L11a15

L11a17

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a16 at Knotilus!

Link Presentations

[edit Notes on L11a16's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{6}-t(2)^{5}+t(2)^{4}-t(2)^{3}+t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{7/2}}}$ (db)
Jones polynomial	$-8q^{9/2}+6q^{7/2}-6q^{5/2}+3q^{3/2}-q^{21/2}+3q^{19/2}-4q^{17/2}+6q^{15/2}-7q^{13/2}+8q^{11/2}-3{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{9}a^{-5}-z^{7}a^{-3}+7z^{7}a^{-5}-z^{7}a^{-7}-5z^{5}a^{-3}+16z^{5}a^{-5}-5z^{5}a^{-7}-5z^{3}a^{-3}+12z^{3}a^{-5}-6z^{3}a^{-7}+3za^{-3}-3za^{-5}+3a^{-3}z^{-1}-5a^{-5}z^{-1}+2a^{-7}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-13}+3z^{4}a^{-12}-2z^{2}a^{-12}+4z^{5}a^{-11}-3z^{3}a^{-11}+4z^{6}a^{-10}-3z^{4}a^{-10}-2z^{2}a^{-10}+a^{-10}+4z^{7}a^{-9}-6z^{5}a^{-9}+4z^{8}a^{-8}-10z^{6}a^{-8}+4z^{4}a^{-8}+4z^{9}a^{-7}-16z^{7}a^{-7}+20z^{5}a^{-7}-12z^{3}a^{-7}+za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}-6z^{8}a^{-6}-z^{6}a^{-6}+6z^{4}a^{-6}+4z^{2}a^{-6}-5a^{-6}+7z^{9}a^{-5}-38z^{7}a^{-5}+63z^{5}a^{-5}-35z^{3}a^{-5}-za^{-5}+5a^{-5}z^{-1}+2z^{10}a^{-4}-9z^{8}a^{-4}+8z^{6}a^{-4}+2z^{4}a^{-4}+3z^{2}a^{-4}-5a^{-4}+3z^{9}a^{-3}-18z^{7}a^{-3}+33z^{5}a^{-3}-19z^{3}a^{-3}-2za^{-3}+3a^{-3}z^{-1}+z^{8}a^{-2}-5z^{6}a^{-2}+6z^{4}a^{-2}-z^{2}a^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 22                       1 1 20                     2   -2 18                   2 1   1 16                 4 2     -2 14               3 2       1 12             5 4         -1 10           3 3           0 8         3 5             2 6       3 3               0 4     2 5                 3 2   1 1                   0 0   2                     2 -2 1                       -1

Integral Khovanov Homology (db, data source)

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

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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