L11a345

L11a344

L11a346

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a345 at Knotilus!

Link Presentations

[edit Notes on L11a345's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X6,17,7,18 X16,7,17,8 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code	{1, -11, 2, -10, 9, -7, 8, -2, 10, -9}, {11, -1, 3, -6, 5, -8, 7, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{3}-4u^{3}v^{2}+6u^{3}v-2u^{3}-2u^{2}v^{3}+6u^{2}v^{2}-6u^{2}v+2u^{2}+2uv^{3}-6uv^{2}+6uv-2u-2v^{3}+6v^{2}-4v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-14q^{5/2}+17q^{3/2}-19{\sqrt {q}}+{\frac {18}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+10az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-8a^{3}z+12az-9za^{-1}+2za^{-3}+2a^{5}z^{-1}-5a^{3}z^{-1}+6az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+4z^{5}a^{-5}+9a^{5}z^{3}-z^{3}a^{-5}-7a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-7a^{4}z^{6}+9z^{6}a^{-4}+7a^{4}z^{4}-8z^{4}a^{-4}-a^{4}z^{2}+3z^{2}a^{-4}-a^{4}-a^{-4}+2a^{3}z^{9}-a^{3}z^{7}+13z^{7}a^{-3}-16a^{3}z^{5}-19z^{5}a^{-3}+31a^{3}z^{3}+11z^{3}a^{-3}-21a^{3}z-4za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+a^{2}z^{10}+6a^{2}z^{8}+11z^{8}a^{-2}-24a^{2}z^{6}-12z^{6}a^{-2}+21a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}+9z^{2}a^{-2}-a^{2}-3a^{-2}+7az^{9}+5z^{9}a^{-1}-6az^{7}+9z^{7}a^{-1}-30az^{5}-42z^{5}a^{-1}+50az^{3}+40z^{3}a^{-1}-29az-19za^{-1}+6az^{-1}+4a^{-1}z^{-1}+z^{10}+15z^{8}-38z^{6}+18z^{4}+4z^{2}-3$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   6 1   -5 6                 8 3     5 4               9 6       -3 2             10 8         2 0           9 10           1 -2         6 9             -3 -4       4 9               5 -6     2 6                 -4 -8   1 5                   4 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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