L11a7

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a7 at Knotilus!

Link Presentations

[edit Notes on L11a7's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(2v^{4}-3v^{3}+3v^{2}-3v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$16q^{9/2}-17q^{7/2}+14q^{5/2}-12q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+3q^{17/2}-5q^{15/2}+10q^{13/2}-14q^{11/2}+7{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-3z^{3}a^{-7}-za^{-7}+a^{-7}z^{-1}+z^{7}a^{-5}+4z^{5}a^{-5}+5z^{3}a^{-5}+za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+z^{3}a^{-3}-za^{-3}-z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-2z^{3}a^{-11}+3z^{6}a^{-10}-7z^{4}a^{-10}+4z^{2}a^{-10}+4z^{7}a^{-9}-7z^{5}a^{-9}+3z^{3}a^{-9}+4z^{8}a^{-8}-4z^{6}a^{-8}-3z^{4}a^{-8}+7z^{2}a^{-8}-2a^{-8}+4z^{9}a^{-7}-8z^{7}a^{-7}+13z^{5}a^{-7}-9z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}-2z^{6}a^{-6}+z^{4}a^{-6}+6z^{2}a^{-6}-5a^{-6}+9z^{9}a^{-5}-27z^{7}a^{-5}+38z^{5}a^{-5}-24z^{3}a^{-5}+3za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+2z^{8}a^{-4}-12z^{6}a^{-4}+8z^{4}a^{-4}+2z^{2}a^{-4}-3a^{-4}+5z^{9}a^{-3}-11z^{7}a^{-3}+6z^{5}a^{-3}-4z^{3}a^{-3}+2za^{-3}+6z^{8}a^{-2}-16z^{6}a^{-2}+9z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-11z^{5}a^{-1}+6z^{3}a^{-1}+za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     2   -2 16                   3 1   2 14                 7 2     -5 12               7 3       4 10             9 7         -2 8           8 7           1 6         6 9             3 4       6 8               -2 2     3 8                 5 0   1 4                   -3 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

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

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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