L11a31

L11a30

L11a32

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a31 at Knotilus!

Link Presentations

[edit Notes on L11a31's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-3t(2)^{5}-4t(1)t(2)^{4}+5t(2)^{4}+5t(1)t(2)^{3}-5t(2)^{3}-5t(1)t(2)^{2}+5t(2)^{2}+5t(1)t(2)-4t(2)-3t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-10q^{11/2}+13q^{9/2}-15q^{7/2}+14q^{5/2}-12q^{3/2}+8{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}-2z^{5}a^{-5}-5z^{3}a^{-5}+2a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}-5za^{-3}-4a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-2za^{-1}+az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-5z^{8}a^{-2}-9z^{8}a^{-4}-6z^{8}a^{-6}-2z^{8}-az^{7}+11z^{7}a^{-1}+22z^{7}a^{-3}+3z^{7}a^{-5}-7z^{7}a^{-7}+30z^{6}a^{-2}+35z^{6}a^{-4}+6z^{6}a^{-6}-6z^{6}a^{-8}+7z^{6}+5az^{5}-13z^{5}a^{-1}-27z^{5}a^{-3}+3z^{5}a^{-5}+9z^{5}a^{-7}-3z^{5}a^{-9}-42z^{4}a^{-2}-44z^{4}a^{-4}+2z^{4}a^{-6}+8z^{4}a^{-8}-z^{4}a^{-10}-5z^{4}-8az^{3}+10z^{3}a^{-1}+27z^{3}a^{-3}+2z^{3}a^{-5}-5z^{3}a^{-7}+2z^{3}a^{-9}+23z^{2}a^{-2}+27z^{2}a^{-4}-4z^{2}a^{-6}-6z^{2}a^{-8}+z^{2}a^{-10}-z^{2}+5az-6za^{-1}-17za^{-3}-5za^{-5}+za^{-7}-5a^{-2}-6a^{-4}+a^{-6}+2a^{-8}+1-az^{-1}+a^{-1}z^{-1}+4a^{-3}z^{-1}+2a^{-5}z^{-1}$ (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 18                       1 -1 16                     2   2 14                   5 1   -4 12                 5 2     3 10               8 5       -3 8             7 5         2 6           7 8           1 4         5 7             -2 2       4 8               4 0     2 4                 -2 -2     4                   4 -4 1 2                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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