L11a10

L11a9

L11a11

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a10 at Knotilus!

Link Presentations

[edit Notes on L11a10's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(4t(2)^{2}-7t(2)+4\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{17/2}+4q^{15/2}-7q^{13/2}+12q^{11/2}-16q^{9/2}+18q^{7/2}-20q^{5/2}+16q^{3/2}-13{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-7}+a^{-7}z^{-1}+z^{5}a^{-5}-2za^{-5}-2a^{-5}z^{-1}+2z^{5}a^{-3}+3z^{3}a^{-3}+2za^{-3}+z^{5}a^{-1}-az^{3}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-9}-3z^{5}a^{-9}+2z^{3}a^{-9}+4z^{8}a^{-8}-15z^{6}a^{-8}+16z^{4}a^{-8}-3z^{2}a^{-8}-2a^{-8}+5z^{9}a^{-7}-16z^{7}a^{-7}+13z^{5}a^{-7}-2z^{3}a^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}+5z^{8}a^{-6}-32z^{6}a^{-6}+32z^{4}a^{-6}-3z^{2}a^{-6}-5a^{-6}+11z^{9}a^{-5}-27z^{7}a^{-5}+11z^{5}a^{-5}+5z^{3}a^{-5}-3za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+11z^{8}a^{-4}-37z^{6}a^{-4}+25z^{4}a^{-4}-z^{2}a^{-4}-3a^{-4}+6z^{9}a^{-3}+z^{7}a^{-3}-24z^{5}a^{-3}+19z^{3}a^{-3}-4za^{-3}+10z^{8}a^{-2}-12z^{6}a^{-2}+a^{2}z^{4}+z^{4}a^{-2}+a^{-2}+11z^{7}a^{-1}+4az^{5}-15z^{5}a^{-1}-2az^{3}+8z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+8z^{6}-7z^{4}+z^{2}$ (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 18                       1 1 16                     3   -3 14                   4 1   3 12                 8 3     -5 10               8 4       4 8             10 8         -2 6           10 8           2 4         6 10             4 2       7 10               -3 0     3 8                 5 -2   1 5                   -4 -4   3                     3 -6 1                       -1

Integral Khovanov Homology (db, data source)

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

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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