L11a204

L11a203

L11a205

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a204 at Knotilus!

Link Presentations

[edit Notes on L11a204's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X22,20,7,19 X20,12,21,11 X12,22,13,21 X6718 X4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 8, -9, 5, -6, 11, -4, 6, -5, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{4}-4u^{2}v^{3}+4u^{2}v^{2}-u^{2}v-uv^{4}+5uv^{3}-7uv^{2}+5uv-u-v^{3}+4v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	$q^{3/2}-2{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {13}{q^{9/2}}}-{\frac {12}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+2za^{7}-z^{7}a^{5}-4z^{5}a^{5}-4z^{3}a^{5}+za^{5}+2a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}-10za^{3}-3a^{3}z^{-1}+z^{5}a+4z^{3}a+4za+az^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+6z^{4}a^{10}-2z^{2}a^{10}-4z^{7}a^{9}+6z^{5}a^{9}-4z^{8}a^{8}+6z^{6}a^{8}-3z^{4}a^{8}+2z^{2}a^{8}-3z^{9}a^{7}+5z^{7}a^{7}-7z^{5}a^{7}+6z^{3}a^{7}-3za^{7}-z^{10}a^{6}-3z^{8}a^{6}+11z^{6}a^{6}-16z^{4}a^{6}+6z^{2}a^{6}-5z^{9}a^{5}+14z^{7}a^{5}-17z^{5}a^{5}+3z^{3}a^{5}+4za^{5}-2a^{5}z^{-1}-z^{10}a^{4}-z^{8}a^{4}+6z^{6}a^{4}-5z^{4}a^{4}-3z^{2}a^{4}+3a^{4}-2z^{9}a^{3}+3z^{7}a^{3}+4z^{5}a^{3}-12z^{3}a^{3}+12za^{3}-3a^{3}z^{-1}-2z^{8}a^{2}+3z^{6}a^{2}+6z^{4}a^{2}-9z^{2}a^{2}+3a^{2}-2z^{7}a+7z^{5}a-7z^{3}a+4za-az^{-1}-z^{6}+4z^{4}-4z^{2}+1$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     1   1 0                   3 1   -2 -2                 5 1     4 -4               6 4       -2 -6             7 4         3 -8           6 6           0 -10         6 7             -1 -12       3 6               3 -14     3 6                 -3 -16   1 4                   3 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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