L11a12

L11a11

L11a13

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a12 at Knotilus!

Link Presentations

[edit Notes on L11a12's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,12,15,11 X8493 X12,5,13,6 X22,13,5,14 X18,15,19,16 X16,9,17,10 X10,17,11,18 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 9, -10, 4, -6, 7, -4, 8, -9, 10, -8, 11, -2, 3, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-5v^{3}+7v^{2}-5v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-5{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {21}{q^{5/2}}}-{\frac {25}{q^{7/2}}}+{\frac {24}{q^{9/2}}}-{\frac {21}{q^{11/2}}}+{\frac {15}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {4}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-3a^{7}z^{3}-4a^{7}z-a^{7}z^{-1}+3a^{5}z^{5}+7a^{5}z^{3}+6a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-4a^{3}z^{3}-2a^{3}z+az^{5}+az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}-a^{11}z^{3}+4a^{10}z^{6}-6a^{10}z^{4}+3a^{10}z^{2}+7a^{9}z^{7}-9a^{9}z^{5}+5a^{9}z^{3}-2a^{9}z+8a^{8}z^{8}-5a^{8}z^{6}-7a^{8}z^{4}+9a^{8}z^{2}-2a^{8}+6a^{7}z^{9}+5a^{7}z^{7}-25a^{7}z^{5}+24a^{7}z^{3}-8a^{7}z+a^{7}z^{-1}+2a^{6}z^{10}+17a^{6}z^{8}-36a^{6}z^{6}+15a^{6}z^{4}+8a^{6}z^{2}-5a^{6}+13a^{5}z^{9}-10a^{5}z^{7}-26a^{5}z^{5}+31a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+2a^{4}z^{10}+18a^{4}z^{8}-47a^{4}z^{6}+27a^{4}z^{4}-3a^{4}+7a^{3}z^{9}-3a^{3}z^{7}-21a^{3}z^{5}+18a^{3}z^{3}-4a^{3}z+9a^{2}z^{8}-19a^{2}z^{6}+10a^{2}z^{4}-2a^{2}z^{2}+a^{2}+5az^{7}-10az^{5}+5az^{3}+az-az^{-1}+z^{6}-z^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     4   4 0                   6 1   -5 -2                 11 4     7 -4               12 8       -4 -6             13 9         4 -8           11 12           1 -10         10 13             -3 -12       5 11               6 -14     3 10                 -7 -16   1 5                   4 -18   3                     -3 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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