L11a151

L11a150

L11a152

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a151 at Knotilus!

Link Presentations

[edit Notes on L11a151's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{3}-4u^{2}v^{2}+4u^{2}v-2u^{2}+2uv^{4}-8uv^{3}+11uv^{2}-8uv+2u-2v^{4}+4v^{3}-4v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	${\frac {17}{q^{9/2}}}-{\frac {16}{q^{7/2}}}+{\frac {11}{q^{5/2}}}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {11}{q^{15/2}}}+{\frac {15}{q^{13/2}}}-{\frac {18}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-za^{9}-a^{9}z^{-1}+z^{5}a^{7}+z^{3}a^{7}+2za^{7}+2a^{7}z^{-1}+2z^{5}a^{5}+3z^{3}a^{5}+za^{5}+z^{5}a^{3}-2za^{3}-a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-2z^{2}a^{12}-3z^{7}a^{11}+8z^{5}a^{11}-5z^{3}a^{11}-5z^{8}a^{10}+14z^{6}a^{10}-14z^{4}a^{10}+8z^{2}a^{10}-2a^{10}-4z^{9}a^{9}+7z^{7}a^{9}-5z^{5}a^{9}+7z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{10}a^{8}-9z^{8}a^{8}+27z^{6}a^{8}-29z^{4}a^{8}+18z^{2}a^{8}-5a^{8}-7z^{9}a^{7}+10z^{7}a^{7}-5z^{5}a^{7}+5z^{3}a^{7}-5za^{7}+2a^{7}z^{-1}-z^{10}a^{6}-9z^{8}a^{6}+19z^{6}a^{6}-15z^{4}a^{6}+7z^{2}a^{6}-3a^{6}-3z^{9}a^{5}-5z^{7}a^{5}+17z^{5}a^{5}-15z^{3}a^{5}+4za^{5}-5z^{8}a^{4}+4z^{6}a^{4}+2z^{4}a^{4}-3z^{2}a^{4}+a^{4}-5z^{7}a^{3}+8z^{5}a^{3}-6z^{3}a^{3}+4za^{3}-a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-2z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     2   -2 -2                   5 1   4 -4                 7 3     -4 -6               9 4       5 -8             9 8         -1 -10           9 8           1 -12         6 9             3 -14       5 9               -4 -16     2 6                 4 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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