L11a216

L11a215

L11a217

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a216 at Knotilus!

Link Presentations

[edit Notes on L11a216's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,13,19,14 X22,19,7,20 X20,9,21,10 X10,21,11,22 X2738 X6,11,1,12
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 8, -9, 11, -2, 6, -3, 5, -4, 3, -6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4u^{2}v^{3}-6u^{2}v^{2}+4u^{2}v-u^{2}+2uv^{4}-8uv^{3}+13uv^{2}-8uv+2u-v^{4}+4v^{3}-6v^{2}+4v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {13}{q^{21/2}}}+{\frac {18}{q^{19/2}}}-{\frac {20}{q^{17/2}}}+{\frac {20}{q^{15/2}}}-{\frac {18}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)+3a^{11}z^{3}+3a^{11}z-a^{11}z^{-1}-2a^{9}z^{5}-2a^{9}z^{3}+3a^{9}z+3a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-7a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-10a^{15}z^{5}+7a^{15}z^{3}-a^{15}z+6a^{14}z^{8}-13a^{14}z^{6}+7a^{14}z^{4}-a^{14}z^{2}+4a^{13}z^{9}+2a^{13}z^{7}-22a^{13}z^{5}+18a^{13}z^{3}-3a^{13}z+a^{12}z^{10}+14a^{12}z^{8}-36a^{12}z^{6}+27a^{12}z^{4}-8a^{12}z^{2}-a^{12}+8a^{11}z^{9}-3a^{11}z^{7}-19a^{11}z^{5}+16a^{11}z^{3}-3a^{11}z+a^{11}z^{-1}+a^{10}z^{10}+14a^{10}z^{8}-29a^{10}z^{6}+17a^{10}z^{4}+a^{10}z^{2}-3a^{10}+4a^{9}z^{9}+5a^{9}z^{7}-19a^{9}z^{5}+19a^{9}z^{3}-10a^{9}z+3a^{9}z^{-1}+6a^{8}z^{8}-4a^{8}z^{6}-5a^{8}z^{4}+8a^{8}z^{2}-3a^{8}+6a^{7}z^{7}-11a^{7}z^{5}+12a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 7 3     -4 -12               11 5       6 -14             9 7         -2 -16           11 11           0 -18         8 10             2 -20       5 10               -5 -22     3 8                 5 -24   1 5                   -4 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-8$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-7$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-6$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=-5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L11a215 L11a217