L11a217

L11a216

L11a218

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a217 at Knotilus!

Link Presentations

[edit Notes on L11a217's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X16,6,17,5 X22,13,7,14 X18,15,19,16 X14,21,15,22 X20,10,21,9 X4,18,5,17 X10,20,11,19 X2738 X6,11,1,12
Gauss code	{1, -10, 2, -8, 3, -11}, {10, -1, 7, -9, 11, -2, 4, -6, 5, -3, 8, -5, 9, -7, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{3}-7u^{2}v^{2}+6u^{2}v-u^{2}+uv^{4}-9uv^{3}+17uv^{2}-9uv+u-v^{4}+6v^{3}-7v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {15}{q^{9/2}}}-q^{7/2}+{\frac {19}{q^{7/2}}}+3q^{5/2}-{\frac {23}{q^{5/2}}}-8q^{3/2}+{\frac {22}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+14{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-3a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}-za^{-3}-az^{5}+3az^{3}+2z^{3}a^{-1}+5az+2az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-6a^{5}z^{9}-12a^{3}z^{9}-6az^{9}-7a^{6}z^{8}-13a^{4}z^{8}-14a^{2}z^{8}-8z^{8}-4a^{7}z^{7}+5a^{5}z^{7}+16a^{3}z^{7}+az^{7}-6z^{7}a^{-1}-a^{8}z^{6}+15a^{6}z^{6}+39a^{4}z^{6}+39a^{2}z^{6}-3z^{6}a^{-2}+13z^{6}+9a^{7}z^{5}+13a^{5}z^{5}+9a^{3}z^{5}+15az^{5}+9z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-8a^{6}z^{4}-30a^{4}z^{4}-37a^{2}z^{4}+4z^{4}a^{-2}-13z^{4}-6a^{7}z^{3}-14a^{5}z^{3}-22a^{3}z^{3}-23az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}+8a^{4}z^{2}+13a^{2}z^{2}-z^{2}a^{-2}+7z^{2}+2a^{7}z+5a^{5}z+10a^{3}z+12az+4za^{-1}-za^{-3}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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                   6 1   5 2                 8 2     -6 0               11 6       5 -2             12 9         -3 -4           11 10           1 -6         9 13             4 -8       6 10               -4 -10     3 9                 6 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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