L11a81

L11a80

L11a82

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a81 at Knotilus!

Link Presentations

[edit Notes on L11a81's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {5uv^{4}-11uv^{3}+12uv^{2}-7uv+2u+2v^{5}-7v^{4}+12v^{3}-11v^{2}+5v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {3}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {14}{q^{21/2}}}+{\frac {20}{q^{19/2}}}-{\frac {24}{q^{17/2}}}+{\frac {24}{q^{15/2}}}-{\frac {22}{q^{13/2}}}+{\frac {16}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+3a^{11}z^{3}+4a^{11}z+a^{11}z^{-1}-2a^{9}z^{5}-a^{9}z^{3}+4a^{9}z+2a^{9}z^{-1}-4a^{7}z^{5}-11a^{7}z^{3}-9a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-a^{5}z^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+3z^{4}a^{16}-3z^{2}a^{16}+a^{16}-3z^{7}a^{15}+7z^{5}a^{15}-5z^{3}a^{15}+za^{15}-5z^{8}a^{14}+9z^{6}a^{14}-4z^{4}a^{14}-5z^{9}a^{13}+4z^{7}a^{13}+5z^{5}a^{13}-4z^{3}a^{13}-za^{13}+a^{13}z^{-1}-2z^{10}a^{12}-12z^{8}a^{12}+34z^{6}a^{12}-31z^{4}a^{12}+15z^{2}a^{12}-3a^{12}-13z^{9}a^{11}+18z^{7}a^{11}-3z^{5}a^{11}+2z^{3}a^{11}-4za^{11}+a^{11}z^{-1}-2z^{10}a^{10}-19z^{8}a^{10}+45z^{6}a^{10}-31z^{4}a^{10}+7z^{2}a^{10}-8z^{9}a^{9}+z^{7}a^{9}+17z^{5}a^{9}-15z^{3}a^{9}+7za^{9}-2a^{9}z^{-1}-12z^{8}a^{8}+17z^{6}a^{8}-4z^{4}a^{8}-5z^{2}a^{8}+3a^{8}-10z^{7}a^{7}+17z^{5}a^{7}-15z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-4z^{6}a^{6}+3z^{4}a^{6}-z^{5}a^{5}+z^{3}a^{5}$ (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                     4 1 -3 -8                   7     7 -10                 9 4     -5 -12               13 7       6 -14             12 10         -2 -16           12 12           0 -18         8 12             4 -20       6 12               -6 -22     2 8                 6 -24   1 6                   -5 -26   2                     2 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-8$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-7$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-6$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-5$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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