L11a116

L11a115

L11a117

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a116 at Knotilus!

Link Presentations

[edit Notes on L11a116's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3t(2)^{3}+6t(1)t(2)^{2}-8t(2)^{2}-8t(1)t(2)+6t(2)+3t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {10}{q^{13/2}}}-{\frac {10}{q^{15/2}}}+{\frac {8}{q^{17/2}}}-{\frac {6}{q^{19/2}}}+{\frac {4}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+3za^{11}+2a^{11}z^{-1}-2z^{3}a^{9}-za^{9}-a^{9}z^{-1}-3z^{3}a^{7}-za^{7}+a^{7}z^{-1}-3z^{3}a^{5}-3za^{5}-a^{5}z^{-1}-z^{3}a^{3}$ (db)
Kauffman polynomial	$a^{14}z^{8}-6a^{14}z^{6}+12a^{14}z^{4}-10a^{14}z^{2}+3a^{14}+2a^{13}z^{9}-11a^{13}z^{7}+19a^{13}z^{5}-12a^{13}z^{3}+3a^{13}z-a^{13}z^{-1}+a^{12}z^{10}-20a^{12}z^{6}+45a^{12}z^{4}-32a^{12}z^{2}+7a^{12}+6a^{11}z^{9}-28a^{11}z^{7}+40a^{11}z^{5}-23a^{11}z^{3}+9a^{11}z-2a^{11}z^{-1}+a^{10}z^{10}+5a^{10}z^{8}-34a^{10}z^{6}+48a^{10}z^{4}-23a^{10}z^{2}+4a^{10}+4a^{9}z^{9}-10a^{9}z^{7}+4a^{9}z^{5}-4a^{9}z^{3}+5a^{9}z-a^{9}z^{-1}+6a^{8}z^{8}-13a^{8}z^{6}+4a^{8}z^{4}-a^{8}z^{2}+7a^{7}z^{7}-11a^{7}z^{5}+2a^{7}z-a^{7}z^{-1}+7a^{6}z^{6}-8a^{6}z^{4}+a^{6}+6a^{5}z^{5}-6a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}+3a^{4}z^{4}+a^{3}z^{3}$ (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 -2                       1 1 -4                     3 1 -2 -6                   3     3 -8                 4 3     -1 -10               6 3       3 -12             5 5         0 -14           5 5           0 -16         3 5             2 -18       3 5               -2 -20     1 3                 2 -22   1 3                   -2 -24   1                     1 -26 1                       -1

Integral Khovanov Homology (db, data source)

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

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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