L11a374

L11a373

L11a375

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a374 at Knotilus!

Link Presentations

[edit Notes on L11a374's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(uv-u+1)(uv-v+1)\left(u^{2}v^{2}+uv+1\right)}{u^{2}v^{2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+6a^{7}z^{5}+12a^{7}z^{3}+9a^{7}z+a^{7}z^{-1}-a^{5}z^{9}-8a^{5}z^{7}-24a^{5}z^{5}-33a^{5}z^{3}-18a^{5}z-a^{5}z^{-1}+a^{3}z^{7}+6a^{3}z^{5}+11a^{3}z^{3}+6a^{3}z$ (db)
Kauffman polynomial	$-z^{3}a^{13}+za^{13}-2z^{4}a^{12}+z^{2}a^{12}-3z^{5}a^{11}+2z^{3}a^{11}-za^{11}-4z^{6}a^{10}+6z^{4}a^{10}-4z^{2}a^{10}-4z^{7}a^{9}+7z^{5}a^{9}-2z^{3}a^{9}-4z^{8}a^{8}+11z^{6}a^{8}-9z^{4}a^{8}+5z^{2}a^{8}-3z^{9}a^{7}+10z^{7}a^{7}-12z^{5}a^{7}+15z^{3}a^{7}-9za^{7}+a^{7}z^{-1}-z^{10}a^{6}-z^{8}a^{6}+18z^{6}a^{6}-29z^{4}a^{6}+16z^{2}a^{6}-a^{6}-5z^{9}a^{5}+26z^{7}a^{5}-46z^{5}a^{5}+40z^{3}a^{5}-18za^{5}+a^{5}z^{-1}-z^{10}a^{4}+2z^{8}a^{4}+9z^{6}a^{4}-23z^{4}a^{4}+12z^{2}a^{4}-2z^{9}a^{3}+12z^{7}a^{3}-24z^{5}a^{3}+20z^{3}a^{3}-7za^{3}-z^{8}a^{2}+6z^{6}a^{2}-11z^{4}a^{2}+6z^{2}a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     1   1 -2                   2 1   -1 -4                 4 1     3 -6               3 3       0 -8             5 3         2 -10           3 3           0 -12         4 5             -1 -14       2 3               1 -16     2 4                 -2 -18   1 3                   2 -20   1                     -1 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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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