L11n177

L11n176

L11n178

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n177 at Knotilus!

Link Presentations

[edit Notes on L11n177's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X5,15,6,14 X18,12,19,11 X10,4,11,3 X7,13,8,12 X13,17,14,16 X17,7,18,22 X15,1,16,6 X4,21,5,22 X2,20,3,19
Gauss code	{1, -11, 5, -10, -3, 9}, {-6, -1, 2, -5, 4, 6, -7, 3, -9, 7, -8, -4, 11, -2, 10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-2u^{2}v^{3}+3u^{2}v^{2}-3u^{2}v+u^{2}-2uv^{4}+5uv^{3}-5uv^{2}+5uv-2u+v^{4}-3v^{3}+3v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$-2q^{15/2}+6q^{13/2}-9q^{11/2}+12q^{9/2}-14q^{7/2}+12q^{5/2}-11q^{3/2}+7{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-za^{-7}-z^{5}a^{-5}-z^{3}a^{-5}+za^{-5}+z^{7}a^{-3}+4z^{5}a^{-3}+5z^{3}a^{-3}+za^{-3}-a^{-3}z^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3z^{3}a^{-9}-za^{-9}+z^{6}a^{-8}+6z^{4}a^{-8}-5z^{2}a^{-8}+4z^{7}a^{-7}-3z^{5}a^{-7}+4z^{3}a^{-7}-za^{-7}+5z^{8}a^{-6}-8z^{6}a^{-6}+10z^{4}a^{-6}-6z^{2}a^{-6}+2z^{9}a^{-5}+6z^{7}a^{-5}-19z^{5}a^{-5}+12z^{3}a^{-5}-za^{-5}+10z^{8}a^{-4}-21z^{6}a^{-4}+8z^{4}a^{-4}+2z^{9}a^{-3}+6z^{7}a^{-3}-27z^{5}a^{-3}+19z^{3}a^{-3}-2za^{-3}-a^{-3}z^{-1}+5z^{8}a^{-2}-11z^{6}a^{-2}+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-11z^{5}a^{-1}+8z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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 χ 16                   2 2 14                 4   -4 12               5 2   3 10             7 4     -3 8           7 5       2 6         6 8         2 4       5 6           -1 2     3 7             4 0   1 4               -3 -2   3                 3 -4 1                   -1

Integral Khovanov Homology (db, data source)

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

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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