L11a246

L11a245

L11a247

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a246 at Knotilus!

Link Presentations

[edit Notes on L11a246's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,6,15,5 X16,7,17,8 X20,15,21,16 X18,14,19,13 X6,22,7,21 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 $u$, $v$, $w$, ...)	${\frac {u^{3}v^{4}-3u^{3}v^{3}+3u^{3}v^{2}-u^{3}v+u^{2}v^{5}-5u^{2}v^{4}+9u^{2}v^{3}-10u^{2}v^{2}+5u^{2}v-u^{2}-uv^{5}+5uv^{4}-10uv^{3}+9uv^{2}-5uv+u-v^{4}+3v^{3}-3v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {15}{q^{9/2}}}-q^{7/2}+{\frac {21}{q^{7/2}}}+5q^{5/2}-{\frac {25}{q^{5/2}}}-11q^{3/2}+{\frac {25}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+17{\sqrt {q}}-{\frac {23}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+2az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+3a^{3}z^{3}-az^{3}-z^{3}a^{-1}-3az+za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+4a^{7}z^{7}-10a^{7}z^{5}+6a^{7}z^{3}+7a^{6}z^{8}-17a^{6}z^{6}+13a^{6}z^{4}-3a^{6}z^{2}+7a^{5}z^{9}-12a^{5}z^{7}+3a^{5}z^{5}+6a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+9a^{4}z^{8}-32a^{4}z^{6}+29a^{4}z^{4}-7a^{4}z^{2}-a^{4}+16a^{3}z^{9}-30a^{3}z^{7}+17a^{3}z^{5}+z^{5}a^{-3}-a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}+15a^{2}z^{8}-36a^{2}z^{6}+5z^{6}a^{-2}+21a^{2}z^{4}-4z^{4}a^{-2}-4a^{2}z^{2}+9az^{9}-3az^{7}+11z^{7}a^{-1}-12az^{5}-15z^{5}a^{-1}+3az^{3}+4z^{3}a^{-1}+2az+za^{-1}+13z^{8}-17z^{6}+3z^{4}$ (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                     4   -4 4                   7 1   6 2                 10 4     -6 0               13 7       6 -2             13 11         -2 -4           12 12           0 -6         9 13             4 -8       6 12               -6 -10     3 10                 7 -12   1 5                   -4 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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