L11a67

L11a66

L11a68

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a67 at Knotilus!

Link Presentations

[edit Notes on L11a67's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-2t(2)^{5}-5t(1)t(2)^{4}+9t(2)^{4}+12t(1)t(2)^{3}-16t(2)^{3}-16t(1)t(2)^{2}+12t(2)^{2}+9t(1)t(2)-5t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {18}{q^{9/2}}}-q^{7/2}+{\frac {24}{q^{7/2}}}+5q^{5/2}-{\frac {29}{q^{5/2}}}-13q^{3/2}+{\frac {30}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+20{\sqrt {q}}-{\frac {26}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}-3z^{5}a^{3}-6z^{3}a^{3}-7za^{3}-4a^{3}z^{-1}+z^{7}a+3z^{5}a+6z^{3}a+5za+3az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{4}z^{10}-3a^{2}z^{10}-7a^{5}z^{9}-19a^{3}z^{9}-12az^{9}-7a^{6}z^{8}-18a^{4}z^{8}-29a^{2}z^{8}-18z^{8}-4a^{7}z^{7}+2a^{5}z^{7}+20a^{3}z^{7}+az^{7}-13z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+50a^{4}z^{6}+70a^{2}z^{6}-5z^{6}a^{-2}+28z^{6}+9a^{7}z^{5}+22a^{5}z^{5}+27a^{3}z^{5}+31az^{5}+16z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-6a^{6}z^{4}-37a^{4}z^{4}-45a^{2}z^{4}+2z^{4}a^{-2}-14z^{4}-8a^{7}z^{3}-31a^{5}z^{3}-45a^{3}z^{3}-29az^{3}-7z^{3}a^{-1}-a^{8}z^{2}+a^{6}z^{2}+9a^{4}z^{2}+11a^{2}z^{2}+4z^{2}+3a^{7}z+16a^{5}z+24a^{3}z+15az+4za^{-1}-a^{6}-2a^{4}-3a^{2}-1-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   9 1   8 2                 11 4     -7 0               15 9       6 -2             16 12         -4 -4           13 14           -1 -6         11 16             5 -8       7 13               -6 -10     3 12                 9 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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