L11n201

L11n200

L11n202

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n201 at Knotilus!

Link Presentations

[edit Notes on L11n201's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X5,14,6,15 X16,7,17,8 X15,21,16,20 X18,14,19,13 X21,6,22,7 X22,18,9,17 X4,19,5,20 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -9, -3, 7, 4, -11}, {10, -1, 11, -2, 6, 3, -5, -4, 8, -6, 9, 5, -7, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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