L11a143

L11a142

L11a144

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a143 at Knotilus!

Link Presentations

[edit Notes on L11a143's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X20,13,21,14 X14,21,15,22 X2,18,3,17
Gauss code	{1, -11, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, 9, -10, 7, -8, 11, -2, 4, -9, 10, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-6t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-4t(2)^{3}+7t(1)^{2}t(2)^{2}-15t(1)t(2)^{2}+7t(2)^{2}-4t(1)^{2}t(2)+11t(1)t(2)-6t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {27}{q^{9/2}}}-{\frac {27}{q^{7/2}}}+{\frac {22}{q^{5/2}}}+q^{3/2}-{\frac {17}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {11}{q^{15/2}}}+{\frac {18}{q^{13/2}}}-{\frac {24}{q^{11/2}}}-4{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+a^{7}z^{3}-a^{7}z-a^{7}z^{-1}-a^{5}z^{7}-2a^{5}z^{5}+a^{5}z^{3}+5a^{5}z+3a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-5a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}+az^{5}+2az^{3}+az$ (db)
Kauffman polynomial	$-z^{5}a^{11}-5z^{6}a^{10}+4z^{4}a^{10}-11z^{7}a^{9}+14z^{5}a^{9}-4z^{3}a^{9}-14z^{8}a^{8}+19z^{6}a^{8}-5z^{4}a^{8}-2z^{2}a^{8}+a^{8}-10z^{9}a^{7}+3z^{7}a^{7}+17z^{5}a^{7}-12z^{3}a^{7}+3za^{7}-a^{7}z^{-1}-3z^{10}a^{6}-20z^{8}a^{6}+47z^{6}a^{6}-24z^{4}a^{6}-3z^{2}a^{6}+3a^{6}-17z^{9}a^{5}+24z^{7}a^{5}+4z^{5}a^{5}-17z^{3}a^{5}+10za^{5}-3a^{5}z^{-1}-3z^{10}a^{4}-13z^{8}a^{4}+37z^{6}a^{4}-23z^{4}a^{4}+3a^{4}-7z^{9}a^{3}+6z^{7}a^{3}+11z^{5}a^{3}-16z^{3}a^{3}+9za^{3}-2a^{3}z^{-1}-7z^{8}a^{2}+13z^{6}a^{2}-6z^{4}a^{2}-4z^{7}a+9z^{5}a-7z^{3}a+2za-z^{6}+2z^{4}-z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   6 1   -5 -2                 11 3     8 -4               12 7       -5 -6             15 10         5 -8           13 13           0 -10         11 14             -3 -12       7 13               6 -14     4 11                 -7 -16   1 7                   6 -18   4                     -4 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-3$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{15}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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