L11a336

L11a335

L11a337

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a336 at Knotilus!

Link Presentations

[edit Notes on L11a336's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X14,3,15,4 X22,11,9,12 X2,15,3,16 X4,19,5,20 X20,5,21,6 X18,21,19,22 X6,13,7,14 X16,7,17,8 X12,17,13,18
Gauss code	{1, -5, 3, -6, 7, -9, 10, -2}, {2, -1, 4, -11, 9, -3, 5, -10, 11, -8, 6, -7, 8, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3u^{3}v^{2}-3u^{3}v+u^{3}+4u^{2}v^{3}-10u^{2}v^{2}+11u^{2}v-4u^{2}-4uv^{3}+11uv^{2}-10uv+4u+v^{3}-3v^{2}+3v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {13}{q^{11/2}}}-{\frac {20}{q^{13/2}}}+{\frac {23}{q^{15/2}}}-{\frac {23}{q^{17/2}}}+{\frac {21}{q^{19/2}}}-{\frac {16}{q^{21/2}}}+{\frac {10}{q^{23/2}}}-{\frac {5}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}+4z^{3}a^{11}+5za^{11}-3z^{5}a^{9}-6z^{3}a^{9}-2za^{9}+a^{9}z^{-1}-3z^{5}a^{7}-7z^{3}a^{7}-5za^{7}-a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$a^{16}z^{6}-a^{16}z^{4}+5a^{15}z^{7}-10a^{15}z^{5}+4a^{15}z^{3}+a^{15}z+9a^{14}z^{8}-20a^{14}z^{6}+11a^{14}z^{4}-2a^{14}z^{2}+7a^{13}z^{9}-5a^{13}z^{7}-17a^{13}z^{5}+14a^{13}z^{3}-2a^{13}z+2a^{12}z^{10}+17a^{12}z^{8}-51a^{12}z^{6}+43a^{12}z^{4}-15a^{12}z^{2}+13a^{11}z^{9}-18a^{11}z^{7}-3a^{11}z^{5}+7a^{11}z^{3}+a^{11}z+2a^{10}z^{10}+15a^{10}z^{8}-40a^{10}z^{6}+36a^{10}z^{4}-11a^{10}z^{2}+6a^{9}z^{9}-2a^{9}z^{7}-7a^{9}z^{5}+9a^{9}z^{3}-3a^{9}z+a^{9}z^{-1}+7a^{8}z^{8}-7a^{8}z^{6}+a^{8}z^{4}+3a^{8}z^{2}-a^{8}+6a^{7}z^{7}-10a^{7}z^{5}+10a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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                 8 3     -5 -12               12 5       7 -14             11 8         -3 -16           12 12           0 -18         9 11             2 -20       7 12               -5 -22     4 10                 6 -24   1 6                   -5 -26   4                     4 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-8$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-7$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-6$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-5$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L11a335 L11a337