L11a36

L11a35

L11a37

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a36 at Knotilus!

Link Presentations

[edit Notes on L11a36's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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