L11a230

L11a229

L11a231

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a230 at Knotilus!

Link Presentations

[edit Notes on L11a230's Link Presentations]

Planar diagram presentation	X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X6718 X4,15,5,16 X22,20,7,19 X20,10,21,9 X10,22,11,21
Gauss code	{1, -2, 3, -8, 4, -7}, {7, -1, 10, -11, 2, -3, 5, -6, 8, -4, 6, -5, 9, -10, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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