L11a227

L11a226

L11a228

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a227 at Knotilus!

Link Presentations

[edit Notes on L11a227's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X4758 X16,5,17,6 X14,18,15,17 X6,15,1,16 X22,14,7,13 X18,22,19,21 X2,11,3,12 X12,3,13,4 X10,19,11,20
Gauss code	{1, -9, 10, -3, 4, -6}, {3, -1, 2, -11, 9, -10, 7, -5, 6, -4, 5, -8, 11, -2, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-5u^{2}v^{3}+8u^{2}v^{2}-3u^{2}v-2uv^{4}+10uv^{3}-15uv^{2}+10uv-2u-3v^{3}+8v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	${\frac {23}{q^{9/2}}}-{\frac {24}{q^{7/2}}}+{\frac {21}{q^{5/2}}}+q^{3/2}-{\frac {16}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {14}{q^{13/2}}}-{\frac {20}{q^{11/2}}}-5{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-5a^{7}z-2a^{7}z^{-1}+3a^{5}z^{5}+7a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}+az^{5}+az^{3}-az$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-6z^{7}a^{9}+9z^{5}a^{9}-8z^{3}a^{9}+5za^{9}-a^{9}z^{-1}-8z^{8}a^{8}+12z^{6}a^{8}-11z^{4}a^{8}+4z^{2}a^{8}-6z^{9}a^{7}-z^{7}a^{7}+19z^{5}a^{7}-26z^{3}a^{7}+12za^{7}-2a^{7}z^{-1}-2z^{10}a^{6}-16z^{8}a^{6}+42z^{6}a^{6}-37z^{4}a^{6}+12z^{2}a^{6}-a^{6}-13z^{9}a^{5}+15z^{7}a^{5}+15z^{5}a^{5}-26z^{3}a^{5}+12za^{5}-2a^{5}z^{-1}-2z^{10}a^{4}-17z^{8}a^{4}+48z^{6}a^{4}-34z^{4}a^{4}+8z^{2}a^{4}-7z^{9}a^{3}+5z^{7}a^{3}+16z^{5}a^{3}-14z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-9z^{8}a^{2}+20z^{6}a^{2}-11z^{4}a^{2}+z^{2}a^{2}-5z^{7}a+10z^{5}a-4z^{3}a-za-z^{6}+z^{4}$ (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                     4   4 0                   6 1   -5 -2                 10 4     6 -4               12 7       -5 -6             12 9         3 -8           11 12           1 -10         9 12             -3 -12       5 11               6 -14     3 9                 -6 -16   1 6                   5 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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