L11a302

L11a301

L11a303

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a302 at Knotilus!

Link Presentations

[edit Notes on L11a302's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2(u-1)(v-1)(uv+1)^{2}}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-7q^{9/2}+8q^{7/2}-{\frac {1}{q^{7/2}}}-10q^{5/2}+{\frac {2}{q^{5/2}}}+10q^{3/2}-{\frac {4}{q^{3/2}}}+q^{15/2}-2q^{13/2}+4q^{11/2}-9{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}-z^{7}a^{-3}+az^{5}-5z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}+4az^{3}-8z^{3}a^{-1}-8z^{3}a^{-3}+4z^{3}a^{-5}+4az-4za^{-1}-4za^{-3}+4za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-4z^{9}a^{-3}-2z^{9}a^{-5}+z^{8}a^{-2}+2z^{8}a^{-4}-2z^{8}a^{-6}-3z^{8}-3az^{7}+4z^{7}a^{-1}+16z^{7}a^{-3}+7z^{7}a^{-5}-2z^{7}a^{-7}-2a^{2}z^{6}+2z^{6}a^{-2}-3z^{6}a^{-4}+5z^{6}a^{-6}-z^{6}a^{-8}+9z^{6}-a^{3}z^{5}+9az^{5}+z^{5}a^{-1}-30z^{5}a^{-3}-14z^{5}a^{-5}+7z^{5}a^{-7}+5a^{2}z^{4}-7z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}a^{-6}+4z^{4}a^{-8}-10z^{4}+3a^{3}z^{3}-9az^{3}-13z^{3}a^{-1}+23z^{3}a^{-3}+18z^{3}a^{-5}-6z^{3}a^{-7}-a^{2}z^{2}+2z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}a^{-8}+2z^{2}-a^{3}z+5az+6za^{-1}-6za^{-3}-5za^{-5}+za^{-7}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     1   1 12                   3 1   -2 10                 4 1     3 8               4 3       -1 6             6 4         2 4           4 4           0 2         5 6             -1 0       3 6               3 -2     1 3                 -2 -4   1 3                   2 -6   1                     -1 -8 1                       1

Integral Khovanov Homology (db, data source)

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

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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