L11a304

L11a303

L11a305

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a304 at Knotilus!

Link Presentations

[edit Notes on L11a304's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,5,17,6 X22,20,9,19 X20,14,21,13 X14,22,15,21 X18,7,19,8 X8,9,1,10 X4,15,5,16 X6,17,7,18
Gauss code	{1, -2, 3, -10, 4, -11, 8, -9}, {9, -1, 2, -3, 6, -7, 10, -4, 11, -8, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(uv+1)\left(u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}+uv^{3}-2uv^{2}+uv+2v^{2}-2v+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+6a^{7}z^{5}+12a^{7}z^{3}+9a^{7}z+2a^{7}z^{-1}-a^{5}z^{9}-8a^{5}z^{7}-24a^{5}z^{5}-33a^{5}z^{3}-19a^{5}z-3a^{5}z^{-1}+a^{3}z^{7}+6a^{3}z^{5}+11a^{3}z^{3}+6a^{3}z+a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{13}z^{3}-a^{13}z+2a^{12}z^{4}-a^{12}z^{2}+3a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+4a^{10}z^{6}-5a^{10}z^{4}+2a^{10}z^{2}+5a^{9}z^{7}-12a^{9}z^{5}+10a^{9}z^{3}-3a^{9}z+4a^{8}z^{8}-9a^{8}z^{6}+2a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{9}-9a^{7}z^{7}+9a^{7}z^{5}-13a^{7}z^{3}+10a^{7}z-2a^{7}z^{-1}+a^{6}z^{10}+a^{6}z^{8}-16a^{6}z^{6}+23a^{6}z^{4}-13a^{6}z^{2}+3a^{6}+5a^{5}z^{9}-26a^{5}z^{7}+48a^{5}z^{5}-45a^{5}z^{3}+21a^{5}z-3a^{5}z^{-1}+a^{4}z^{10}-2a^{4}z^{8}-9a^{4}z^{6}+25a^{4}z^{4}-17a^{4}z^{2}+3a^{4}+2a^{3}z^{9}-12a^{3}z^{7}+24a^{3}z^{5}-19a^{3}z^{3}+6a^{3}z-a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+11a^{2}z^{4}-6a^{2}z^{2}+a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     1   1 -2                   2 1   -1 -4                 4 1     3 -6               4 3       -1 -8             4 3         1 -10           4 4           0 -12         4 4             0 -14       2 4               2 -16     2 4                 -2 -18     2                   2 -20 1 2                     -1 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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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