L11a109

L11a108

L11a110

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a109 at Knotilus!

Link Presentations

[edit Notes on L11a109's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X22,15,5,16 X16,7,17,8 X8,21,9,22 X20,11,21,12 X18,9,19,10 X10,19,11,20 X12,17,13,18 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 7, -8, 6, -9, 11, -2, 3, -4, 9, -7, 8, -6, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4uv^{4}-7uv^{3}+8uv^{2}-6uv+2u+2v^{5}-6v^{4}+8v^{3}-7v^{2}+4v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {3}{q^{25/2}}}+{\frac {7}{q^{23/2}}}-{\frac {11}{q^{21/2}}}+{\frac {15}{q^{19/2}}}-{\frac {17}{q^{17/2}}}+{\frac {17}{q^{15/2}}}-{\frac {16}{q^{13/2}}}+{\frac {10}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+3a^{11}z^{3}+5a^{11}z+a^{11}z^{-1}-2a^{9}z^{5}-3a^{9}z^{3}+2a^{9}z+2a^{9}z^{-1}-3a^{7}z^{5}-9a^{7}z^{3}-8a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{16}z^{6}-3a^{16}z^{4}+3a^{16}z^{2}-a^{16}+3a^{15}z^{7}-8a^{15}z^{5}+6a^{15}z^{3}-a^{15}z+4a^{14}z^{8}-7a^{14}z^{6}-a^{14}z^{4}+3a^{14}z^{2}+3a^{13}z^{9}-11a^{13}z^{5}+6a^{13}z^{3}-a^{13}z^{-1}+a^{12}z^{10}+8a^{12}z^{8}-22a^{12}z^{6}+20a^{12}z^{4}-13a^{12}z^{2}+3a^{12}+7a^{11}z^{9}-11a^{11}z^{7}+6a^{11}z^{5}-4a^{11}z^{3}+3a^{11}z-a^{11}z^{-1}+a^{10}z^{10}+10a^{10}z^{8}-28a^{10}z^{6}+30a^{10}z^{4}-10a^{10}z^{2}+4a^{9}z^{9}-2a^{9}z^{7}-7a^{9}z^{5}+14a^{9}z^{3}-6a^{9}z+2a^{9}z^{-1}+6a^{8}z^{8}-11a^{8}z^{6}+7a^{8}z^{4}+3a^{8}z^{2}-3a^{8}+6a^{7}z^{7}-15a^{7}z^{5}+16a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   4     4 -10                 6 3     -3 -12               10 4       6 -14             8 7         -1 -16           9 9           0 -18         6 8             2 -20       5 9               -4 -22     2 6                 4 -24   1 5                   -4 -26   2                     2 -28 1                       -1

Integral Khovanov Homology (db, data source)

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

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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