L11a185

L11a184

L11a186

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a185 at Knotilus!

Link Presentations

[edit Notes on L11a185's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{3}-6u^{2}v^{2}+4u^{2}v-u^{2}+2uv^{4}-8uv^{3}+11uv^{2}-8uv+2u-v^{4}+4v^{3}-6v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	${\frac {18}{q^{9/2}}}-{\frac {16}{q^{7/2}}}+{\frac {11}{q^{5/2}}}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {4}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {12}{q^{15/2}}}+{\frac {16}{q^{13/2}}}-{\frac {18}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)+a^{9}z^{-1}+a^{7}z^{5}-2a^{7}z-2a^{7}z^{-1}+2a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-2a^{3}z-a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$-z^{6}a^{12}+2z^{4}a^{12}-4z^{7}a^{11}+11z^{5}a^{11}-7z^{3}a^{11}-6z^{8}a^{10}+16z^{6}a^{10}-12z^{4}a^{10}+3z^{2}a^{10}-4z^{9}a^{9}+2z^{7}a^{9}+15z^{5}a^{9}-18z^{3}a^{9}+7za^{9}-a^{9}z^{-1}-z^{10}a^{8}-11z^{8}a^{8}+32z^{6}a^{8}-28z^{4}a^{8}+9z^{2}a^{8}-7z^{9}a^{7}+5z^{7}a^{7}+16z^{5}a^{7}-25z^{3}a^{7}+12za^{7}-2a^{7}z^{-1}-z^{10}a^{6}-10z^{8}a^{6}+22z^{6}a^{6}-18z^{4}a^{6}+7z^{2}a^{6}-a^{6}-3z^{9}a^{5}-6z^{7}a^{5}+21z^{5}a^{5}-22z^{3}a^{5}+10za^{5}-2a^{5}z^{-1}-5z^{8}a^{4}+4z^{6}a^{4}+z^{4}a^{4}-z^{2}a^{4}-5z^{7}a^{3}+8z^{5}a^{3}-6z^{3}a^{3}+4za^{3}-a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-2z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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 7         -2 -10           9 9           0 -12         7 9             2 -14       5 9               -4 -16     3 8                 5 -18   1 4                   -3 -20   3                     3 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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