L11a147

L11a146

L11a148

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a147 at Knotilus!

Link Presentations

[edit Notes on L11a147's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X18,12,19,11 X12,6,13,5 X4,19,5,20 X16,7,17,8 X20,13,21,14 X22,15,7,16 X14,21,15,22 X6,18,1,17
Gauss code	{1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -10, 9, -7, 11, -4, 6, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{6}-2u^{2}v^{5}+2u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{6}+3uv^{5}-3uv^{4}+3uv^{3}-3uv^{2}+3uv-u-v^{5}+2v^{4}-2v^{3}+2v^{2}-2v+1}{uv^{3}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {9}{q^{7/2}}}-{\frac {8}{q^{5/2}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {3}{q^{19/2}}}-{\frac {5}{q^{17/2}}}+{\frac {8}{q^{15/2}}}-{\frac {10}{q^{13/2}}}+{\frac {11}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{5}z^{9}+a^{7}z^{7}-7a^{5}z^{7}+a^{3}z^{7}+5a^{7}z^{5}-17a^{5}z^{5}+5a^{3}z^{5}+7a^{7}z^{3}-16a^{5}z^{3}+6a^{3}z^{3}+2a^{7}z-2a^{5}z-a^{3}z-a^{7}z^{-1}+3a^{5}z^{-1}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}-3z^{4}a^{12}+z^{2}a^{12}-5z^{5}a^{11}+3z^{3}a^{11}-7z^{6}a^{10}+9z^{4}a^{10}-2z^{2}a^{10}-8z^{7}a^{9}+16z^{5}a^{9}-7z^{3}a^{9}+za^{9}-7z^{8}a^{8}+16z^{6}a^{8}-4z^{4}a^{8}-3z^{2}a^{8}+a^{8}-5z^{9}a^{7}+13z^{7}a^{7}-4z^{5}a^{7}-z^{3}a^{7}-a^{7}z^{-1}-2z^{10}a^{6}+z^{8}a^{6}+17z^{6}a^{6}-20z^{4}a^{6}+z^{2}a^{6}+3a^{6}-8z^{9}a^{5}+38z^{7}a^{5}-56z^{5}a^{5}+29z^{3}a^{5}-za^{5}-3a^{5}z^{-1}-2z^{10}a^{4}+7z^{8}a^{4}-z^{6}a^{4}-11z^{4}a^{4}+3z^{2}a^{4}+3a^{4}-3z^{9}a^{3}+17z^{7}a^{3}-31z^{5}a^{3}+19z^{3}a^{3}-2a^{3}z^{-1}-z^{8}a^{2}+5z^{6}a^{2}-7z^{4}a^{2}+2z^{2}a^{2}$ (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                     2   2 -2                   2 1   -1 -4                 6 2     4 -6               4 3       -1 -8             7 5         2 -10           5 5           0 -12         5 6             -1 -14       3 5               2 -16     2 5                 -3 -18   1 3                   2 -20   2                     -2 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a146 L11a148