L11a100

L11a99

L11a101

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a100 at Knotilus!

Link Presentations

[edit Notes on L11a100's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X18,9,19,10 X20,11,21,12 X8,17,9,18 X10,19,11,20 X4,21,1,22 X14,6,15,5 X12,4,13,3 X22,14,5,13 X2,16,3,15
Gauss code	{1, -11, 9, -7}, {8, -1, 2, -5, 3, -6, 4, -9, 10, -8, 11, -2, 5, -3, 6, -4, 7, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)^{2}\left(v^{2}+v+1\right)}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{9}-a^{5}z^{7}+7a^{3}z^{7}-az^{7}-5a^{5}z^{5}+17a^{3}z^{5}-5az^{5}-7a^{5}z^{3}+17a^{3}z^{3}-7az^{3}-3a^{5}z+7a^{3}z-4az-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-4a^{9}z^{3}+4a^{8}z^{6}-5a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-5a^{7}z^{5}+2a^{7}z^{3}+4a^{6}z^{8}-8a^{6}z^{6}+7a^{6}z^{4}-2a^{6}z^{2}+a^{6}+4a^{5}z^{9}-14a^{5}z^{7}+23a^{5}z^{5}-18a^{5}z^{3}+6a^{5}z-a^{5}z^{-1}+2a^{4}z^{10}-4a^{4}z^{8}-3a^{4}z^{6}+11a^{4}z^{4}-8a^{4}z^{2}+3a^{4}+8a^{3}z^{9}-38a^{3}z^{7}+63a^{3}z^{5}-46a^{3}z^{3}+13a^{3}z-3a^{3}z^{-1}+2a^{2}z^{10}-5a^{2}z^{8}-4a^{2}z^{6}+11a^{2}z^{4}-6a^{2}z^{2}+3a^{2}+4az^{9}-19az^{7}+z^{7}a^{-1}+28az^{5}-4z^{5}a^{-1}-19az^{3}+3z^{3}a^{-1}+7az-2az^{-1}+3z^{8}-13z^{6}+13z^{4}-2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   3 1   2 0                 3 2     -1 -2               7 3       4 -4             5 5         0 -6           6 5           1 -8         4 5             1 -10       3 6               -3 -12     2 4                 2 -14   1 3                   -2 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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