L11a103

L11a102

L11a104

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a103 at Knotilus!

Link Presentations

[edit Notes on L11a103's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,6,15,5 X12,4,13,3 X22,14,5,13 X18,9,19,10 X16,11,17,12 X2,16,3,15 X10,17,11,18 X8,19,9,20
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -11, 7, -10, 8, -5, 6, -4, 9, -8, 10, -7, 11, -2, 3, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-6q^{5/2}+7q^{3/2}-10{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{5}-za^{5}+a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}-2a^{3}z^{-1}+z^{5}a+z^{3}a+az^{-1}+z^{5}a^{-1}+2z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}-z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-3a^{5}z^{9}-8a^{3}z^{9}-5az^{9}-a^{6}z^{8}+7a^{4}z^{8}+a^{2}z^{8}-7z^{8}+17a^{5}z^{7}+38a^{3}z^{7}+14az^{7}-7z^{7}a^{-1}+5a^{6}z^{6}+20a^{2}z^{6}-7z^{6}a^{-2}+18z^{6}-30a^{5}z^{5}-51a^{3}z^{5}-5az^{5}+10z^{5}a^{-1}-6z^{5}a^{-3}-7a^{6}z^{4}-13a^{4}z^{4}-29a^{2}z^{4}+8z^{4}a^{-2}-3z^{4}a^{-4}-12z^{4}+17a^{5}z^{3}+15a^{3}z^{3}-6az^{3}+3z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+3a^{6}z^{2}+6a^{4}z^{2}+10a^{2}z^{2}+7z^{2}+a^{5}z+7a^{3}z+6az-3za^{-1}-3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-2a^{3}z^{-1}-az^{-1}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                       1 1 8                     2   -2 6                   4 1   3 4                 3 2     -1 2               7 4       3 0             6 5         -1 -2           4 5           -1 -4         5 6             1 -6       2 4               -2 -8     2 5                 3 -10   1 2                   -1 -12   2                     2 -14 1                       -1

Integral Khovanov Homology (db, data source)

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

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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