L11a58

L11a57

L11a59

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a58 at Knotilus!

Link Presentations

[edit Notes on L11a58's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+6t(2)^{4}+9t(1)t(2)^{3}-11t(2)^{3}-11t(1)t(2)^{2}+9t(2)^{2}+6t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {17}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-11za^{3}-4a^{3}z^{-1}+z^{7}a+4z^{5}a+8z^{3}a+6za+az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z+a^{7}z^{-1}+4a^{6}z^{8}-5a^{6}z^{6}-5a^{6}z^{4}+10a^{6}z^{2}-4a^{6}+3a^{5}z^{9}+5a^{5}z^{7}-25a^{5}z^{5}+27a^{5}z^{3}-14a^{5}z+4a^{5}z^{-1}+a^{4}z^{10}+12a^{4}z^{8}-24a^{4}z^{6}+5a^{4}z^{4}+12a^{4}z^{2}-7a^{4}+8a^{3}z^{9}+a^{3}z^{7}-30a^{3}z^{5}+z^{5}a^{-3}+30a^{3}z^{3}-z^{3}a^{-3}-13a^{3}z+4a^{3}z^{-1}+a^{2}z^{10}+17a^{2}z^{8}-33a^{2}z^{6}+4z^{6}a^{-2}+15a^{2}z^{4}-5z^{4}a^{-2}+3a^{2}z^{2}+2z^{2}a^{-2}-4a^{2}+5az^{9}+7az^{7}+8z^{7}a^{-1}-26az^{5}-12z^{5}a^{-1}+19az^{3}+7z^{3}a^{-1}-5az-2za^{-1}+az^{-1}+9z^{8}-11z^{6}+3z^{4}-1$ (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                     3   -3 4                   6 1   5 2                 9 3     -6 0               10 6       4 -2             12 10         -2 -4           10 9           1 -6         7 12             5 -8       6 10               -4 -10     2 8                 6 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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