L11a102

L11a101

L11a103

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a102 at Knotilus!

Link Presentations

[edit Notes on L11a102's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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