L11a303

L11a302

L11a304

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a303 at Knotilus!

Link Presentations

[edit Notes on L11a303's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{5}-2u^{3}v^{4}+2u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+5u^{2}v^{4}-8u^{2}v^{3}+7u^{2}v^{2}-2u^{2}v-2uv^{4}+7uv^{3}-8uv^{2}+5uv-u-v^{3}+2v^{2}-2v+1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+7q^{5/2}-12q^{3/2}+16{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {18}{q^{3/2}}}-{\frac {17}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+5a^{3}z^{5}+9a^{3}z^{3}+6a^{3}z+2a^{3}z^{-1}-az^{9}-7az^{7}+z^{7}a^{-1}-19az^{5}+5z^{5}a^{-1}-24az^{3}+9z^{3}a^{-1}-14az+6za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-10az^{9}-5z^{9}a^{-1}-6a^{4}z^{8}-5a^{2}z^{8}-5z^{8}a^{-2}-4z^{8}-5a^{5}z^{7}+6a^{3}z^{7}+26az^{7}+12z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+8a^{4}z^{6}+14a^{2}z^{6}+13z^{6}a^{-2}-z^{6}a^{-4}+17z^{6}-a^{7}z^{5}+7a^{5}z^{5}-7a^{3}z^{5}-38az^{5}-15z^{5}a^{-1}+8z^{5}a^{-3}+5a^{6}z^{4}-4a^{4}z^{4}-15a^{2}z^{4}-11z^{4}a^{-2}+3z^{4}a^{-4}-20z^{4}+2a^{7}z^{3}-3a^{5}z^{3}+9a^{3}z^{3}+35az^{3}+16z^{3}a^{-1}-5z^{3}a^{-3}-2a^{6}z^{2}+9a^{2}z^{2}+5z^{2}a^{-2}-2z^{2}a^{-4}+14z^{2}-a^{7}z+a^{5}z-7a^{3}z-16az-7za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   5 1   -4 4                 7 2     5 2               9 5       -4 0             10 7         3 -2           9 10           1 -4         8 9             -1 -6       4 9               5 -8     3 8                 -5 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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