L11a241

L11a240

L11a242

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a241 at Knotilus!

Link Presentations

[edit Notes on L11a241's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{2}-6u^{2}v+5u^{2}-6uv^{2}+13uv-6u+5v^{2}-6v+2}{uv}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {6}{q^{9/2}}}-4q^{7/2}+{\frac {9}{q^{7/2}}}+7q^{5/2}-{\frac {14}{q^{5/2}}}-11q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+15{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3za^{5}-a^{5}z^{-1}+3z^{3}a^{3}+za^{3}-z^{5}a+z^{3}a+za-z^{5}a^{-1}-z^{3}a^{-1}-2za^{-1}+z^{3}a^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-7az^{9}-4z^{9}a^{-1}-3a^{4}z^{8}-7a^{2}z^{8}-6z^{8}a^{-2}-10z^{8}-3a^{5}z^{7}+3a^{3}z^{7}+14az^{7}+4z^{7}a^{-1}-4z^{7}a^{-3}-2a^{6}z^{6}+20a^{2}z^{6}+17z^{6}a^{-2}-z^{6}a^{-4}+36z^{6}-a^{7}z^{5}+3a^{5}z^{5}-7a^{3}z^{5}-11az^{5}+11z^{5}a^{-1}+11z^{5}a^{-3}+3a^{6}z^{4}+5a^{4}z^{4}-24a^{2}z^{4}-13z^{4}a^{-2}+2z^{4}a^{-4}-41z^{4}+3a^{7}z^{3}+a^{5}z^{3}+9a^{3}z^{3}+5az^{3}-12z^{3}a^{-1}-6z^{3}a^{-3}-3a^{4}z^{2}+10a^{2}z^{2}+4z^{2}a^{-2}+17z^{2}-3a^{7}z-2a^{5}z-3a^{3}z-3az+za^{-1}-a^{6}+a^{7}z^{-1}+a^{5}z^{-1}$ (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 10                       1 -1 8                     3   3 6                   4 1   -3 4                 7 3     4 2               8 4       -4 0             8 7         1 -2           9 9           0 -4         5 7             -2 -6       4 9               5 -8     2 5                 -3 -10     4                   4 -12 1 2                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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