L11a62

L11a61

L11a63

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a62 at Knotilus!

Link Presentations

[edit Notes on L11a62's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X22,13,5,14 X14,7,15,8 X20,18,21,17 X10,15,11,16 X8,19,9,20 X18,9,19,10 X16,22,17,21 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 8, -6, 11, -2, 3, -4, 6, -9, 5, -8, 7, -5, 9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}+2t(1)t(2)^{4}-6t(2)^{4}-10t(1)t(2)^{3}+12t(2)^{3}+12t(1)t(2)^{2}-10t(2)^{2}-6t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {13}{q^{5/2}}}-{\frac {18}{q^{7/2}}}+{\frac {20}{q^{9/2}}}-{\frac {20}{q^{11/2}}}+{\frac {16}{q^{13/2}}}-{\frac {12}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-5a^{7}z^{3}-7a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+3a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}-az^{3}$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-8a^{11}z^{5}+7a^{11}z^{3}-3a^{11}z+a^{11}z^{-1}+4a^{10}z^{8}-6a^{10}z^{6}-3a^{10}z^{4}+6a^{10}z^{2}-2a^{10}+3a^{9}z^{9}+3a^{9}z^{7}-21a^{9}z^{5}+23a^{9}z^{3}-13a^{9}z+3a^{9}z^{-1}+a^{8}z^{10}+11a^{8}z^{8}-28a^{8}z^{6}+21a^{8}z^{4}-6a^{8}z^{2}+8a^{7}z^{9}-5a^{7}z^{7}-20a^{7}z^{5}+31a^{7}z^{3}-16a^{7}z+3a^{7}z^{-1}+a^{6}z^{10}+16a^{6}z^{8}-41a^{6}z^{6}+38a^{6}z^{4}-13a^{6}z^{2}+2a^{6}+5a^{5}z^{9}+3a^{5}z^{7}-22a^{5}z^{5}+24a^{5}z^{3}-10a^{5}z+2a^{5}z^{-1}+9a^{4}z^{8}-16a^{4}z^{6}+12a^{4}z^{4}-4a^{4}z^{2}+8a^{3}z^{7}-14a^{3}z^{5}+8a^{3}z^{3}-4a^{3}z+a^{3}z^{-1}+4a^{2}z^{6}-5a^{2}z^{4}+az^{5}-az^{3}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   6 1   5 -4                 8 4     -4 -6               10 5       5 -8             10 8         -2 -10           10 10           0 -12         7 11             4 -14       5 9               -4 -16     2 7                 5 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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