L11a181

L11a180

L11a182

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a181 at Knotilus!

Link Presentations

[edit Notes on L11a181's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X14,5,15,6 X16,8,17,7 X10,4,11,3 X22,14,7,13 X18,12,19,11 X12,18,13,17 X6,15,1,16 X4,21,5,22 X2,20,3,19
Gauss code	{1, -11, 5, -10, 3, -9}, {4, -1, 2, -5, 7, -8, 6, -3, 9, -4, 8, -7, 11, -2, 10, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-7u^{2}v^{3}+8u^{2}v^{2}-3u^{2}v-2uv^{4}+9uv^{3}-15uv^{2}+9uv-2u-3v^{3}+8v^{2}-7v+2}{uv^{2}}}$ (db)
Jones polynomial	$-q^{11/2}+5q^{9/2}-10q^{7/2}+16q^{5/2}-22q^{3/2}+24{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+2z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+4az^{3}-2z^{3}a^{-1}-z^{3}a^{-3}-a^{3}z+4az-6za^{-1}+2za^{-3}+2az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-4z^{10}a^{-2}-4z^{10}-12az^{9}-20z^{9}a^{-1}-8z^{9}a^{-3}-17a^{2}z^{8}-4z^{8}a^{-2}-5z^{8}a^{-4}-16z^{8}-15a^{3}z^{7}+12az^{7}+52z^{7}a^{-1}+24z^{7}a^{-3}-z^{7}a^{-5}-9a^{4}z^{6}+29a^{2}z^{6}+39z^{6}a^{-2}+15z^{6}a^{-4}+62z^{6}-4a^{5}z^{5}+21a^{3}z^{5}+18az^{5}-28z^{5}a^{-1}-19z^{5}a^{-3}+2z^{5}a^{-5}-a^{6}z^{4}+6a^{4}z^{4}-15a^{2}z^{4}-36z^{4}a^{-2}-12z^{4}a^{-4}-46z^{4}+a^{5}z^{3}-12a^{3}z^{3}-22az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}+2a^{2}z^{2}+3z^{2}a^{-2}+6z^{2}+2a^{3}z+9az+10za^{-1}+3za^{-3}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     4   -4 8                   6 1   5 6                 10 4     -6 4               12 6       6 2             12 10         -2 0           13 12           1 -2         9 13             4 -4       7 12               -5 -6     3 10                 7 -8   1 6                   -5 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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