L11a150

L11a149

L11a151

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a150 at Knotilus!

Link Presentations

[edit Notes on L11a150's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-7u^{2}v^{3}+9u^{2}v^{2}-5u^{2}v+u^{2}-3uv^{4}+12uv^{3}-19uv^{2}+12uv-3u+v^{4}-5v^{3}+9v^{2}-7v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {20}{q^{9/2}}}-q^{7/2}+{\frac {27}{q^{7/2}}}+5q^{5/2}-{\frac {32}{q^{5/2}}}-12q^{3/2}+{\frac {31}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {5}{q^{13/2}}}+{\frac {12}{q^{11/2}}}+20{\sqrt {q}}-{\frac {28}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-a^{5}z^{3}+a^{3}z^{7}+2a^{3}z^{5}+a^{3}z^{3}+a^{3}z^{-1}+az^{7}+2az^{5}-z^{5}a^{-1}+az^{3}-z^{3}a^{-1}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-a^{8}z^{4}+5a^{7}z^{7}-8a^{7}z^{5}+3a^{7}z^{3}+11a^{6}z^{8}-22a^{6}z^{6}+13a^{6}z^{4}-2a^{6}z^{2}+12a^{5}z^{9}-19a^{5}z^{7}+4a^{5}z^{5}+2a^{5}z^{3}+5a^{4}z^{10}+16a^{4}z^{8}-53a^{4}z^{6}+36a^{4}z^{4}-6a^{4}z^{2}+26a^{3}z^{9}-47a^{3}z^{7}+18a^{3}z^{5}+z^{5}a^{-3}+a^{3}z-a^{3}z^{-1}+5a^{2}z^{10}+22a^{2}z^{8}-59a^{2}z^{6}+5z^{6}a^{-2}+36a^{2}z^{4}-3z^{4}a^{-2}-6a^{2}z^{2}+a^{2}+14az^{9}-11az^{7}+12z^{7}a^{-1}-9az^{5}-14z^{5}a^{-1}+6az^{3}+5z^{3}a^{-1}+az-az^{-1}+17z^{8}-24z^{6}+11z^{4}-2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   8 1   7 2                 12 4     -8 0               16 8       8 -2             16 13         -3 -4           16 15           1 -6         12 17             5 -8       8 15               -7 -10     4 12                 8 -12   1 8                   -7 -14   4                     4 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-2$ ${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{16}$ $r=-1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{16}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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