L11a338

L11a337

L11a339

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a338 at Knotilus!

Link Presentations

[edit Notes on L11a338's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{3}v^{3}-3u^{3}v^{2}+u^{3}v-4u^{2}v^{3}+12u^{2}v^{2}-9u^{2}v+2u^{2}+2uv^{3}-9uv^{2}+12uv-4u+v^{2}-3v+2}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {21}{q^{9/2}}}-{\frac {22}{q^{7/2}}}+{\frac {18}{q^{5/2}}}+q^{3/2}-{\frac {15}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {13}{q^{13/2}}}-{\frac {18}{q^{11/2}}}-4{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+3za^{7}-z^{7}a^{5}-4z^{5}a^{5}-7z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-3z^{3}a^{3}-2za^{3}-a^{3}z^{-1}+z^{5}a+2z^{3}a+za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-5a^{10}z^{4}+2a^{10}z^{2}+5a^{9}z^{7}-6a^{9}z^{5}+a^{9}z^{3}+7a^{8}z^{8}-12a^{8}z^{6}+13a^{8}z^{4}-7a^{8}z^{2}+6a^{7}z^{9}-7a^{7}z^{7}+4a^{7}z^{5}+a^{7}z+2a^{6}z^{10}+12a^{6}z^{8}-38a^{6}z^{6}+42a^{6}z^{4}-15a^{6}z^{2}+12a^{5}z^{9}-22a^{5}z^{7}+9a^{5}z^{5}+2a^{5}z^{3}+a^{5}z^{-1}+2a^{4}z^{10}+12a^{4}z^{8}-39a^{4}z^{6}+33a^{4}z^{4}-9a^{4}z^{2}-a^{4}+6a^{3}z^{9}-6a^{3}z^{7}-11a^{3}z^{5}+11a^{3}z^{3}-4a^{3}z+a^{3}z^{-1}+7a^{2}z^{8}-15a^{2}z^{6}+7a^{2}z^{4}-2a^{2}z^{2}+4az^{7}-9az^{5}+6az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   6 1   -5 -2                 9 3     6 -4               10 7       -3 -6             12 8         4 -8           9 10           1 -10         9 12             -3 -12       5 10               5 -14     2 8                 -6 -16   1 5                   4 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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