L11a101

L11a100

L11a102

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a101 at Knotilus!

Link Presentations

[edit Notes on L11a101's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)\left(2v^{2}-v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+13{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {19}{q^{3/2}}}-{\frac {19}{q^{5/2}}}+{\frac {16}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+6a^{3}z^{3}+2az^{3}-2z^{3}a^{-1}-2a^{5}z+3a^{3}z-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}+4a^{6}z^{8}-9a^{6}z^{6}+5a^{6}z^{4}-a^{6}z^{2}+4a^{5}z^{9}-8a^{5}z^{7}+9a^{5}z^{5}-10a^{5}z^{3}+4a^{5}z+2a^{4}z^{10}+2a^{4}z^{8}-8a^{4}z^{6}+7a^{4}z^{4}-4a^{4}z^{2}+10a^{3}z^{9}-25a^{3}z^{7}+34a^{3}z^{5}+z^{5}a^{-3}-25a^{3}z^{3}-z^{3}a^{-3}+6a^{3}z+2a^{2}z^{10}+7a^{2}z^{8}-18a^{2}z^{6}+4z^{6}a^{-2}+14a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}+6az^{9}-6az^{7}+8z^{7}a^{-1}+az^{5}-14z^{5}a^{-1}+7z^{3}a^{-1}-2za^{-1}+az^{-1}+a^{-1}z^{-1}+9z^{8}-16z^{6}+10z^{4}-2z^{2}-1$ (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                     3   -3 4                   6 1   5 2                 7 3     -4 0               11 6       5 -2             10 9         -1 -4           9 9           0 -6         7 10             3 -8       4 9               -5 -10     2 7                 5 -12   1 4                   -3 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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