L11a192

L11a191

L11a193

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a192 at Knotilus!

Link Presentations

[edit Notes on L11a192's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X16,6,17,5 X22,11,7,12 X20,13,21,14 X18,15,19,16 X14,19,15,20 X12,21,13,22 X4,18,5,17 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 11, -2, 4, -8, 5, -7, 6, -3, 9, -6, 7, -5, 8, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3t(2)t(1)^{2}-3t(1)^{2}+3t(2)^{2}t(1)-7t(2)t(1)+3t(1)-3t(2)^{2}+3t(2)}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {7}{q^{7/2}}}-{\frac {7}{q^{5/2}}}-q^{3/2}+{\frac {5}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {2}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {5}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{9}+z^{3}a^{7}+2z^{3}a^{5}+za^{5}+2z^{3}a^{3}+za^{3}+a^{3}z^{-1}+z^{3}a-za-az^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{10}+6z^{6}a^{10}-11z^{4}a^{10}+6z^{2}a^{10}-2z^{9}a^{9}+12z^{7}a^{9}-23z^{5}a^{9}+16z^{3}a^{9}-3za^{9}-z^{10}a^{8}+3z^{8}a^{8}+3z^{6}a^{8}-11z^{4}a^{8}+4z^{2}a^{8}-4z^{9}a^{7}+19z^{7}a^{7}-26z^{5}a^{7}+12z^{3}a^{7}-2za^{7}-z^{10}a^{6}+z^{8}a^{6}+7z^{6}a^{6}-8z^{4}a^{6}+z^{2}a^{6}-2z^{9}a^{5}+4z^{7}a^{5}+3z^{5}a^{5}-4z^{3}a^{5}+za^{5}-3z^{8}a^{4}+7z^{6}a^{4}-4z^{4}a^{4}+3z^{2}a^{4}-3z^{7}a^{3}+3z^{5}a^{3}+4z^{3}a^{3}-4za^{3}+a^{3}z^{-1}-3z^{6}a^{2}+2z^{4}a^{2}+z^{2}a^{2}-a^{2}-3z^{5}a+3z^{3}a-3za+az^{-1}-2z^{4}+z^{2}-z^{3}a^{-1}+za^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                       1 1 2                     1   -1 0                   3 1   2 -2                 3 2     -1 -4               4 2       2 -6             4 4         0 -8           3 3           0 -10         3 4             1 -12       2 3               -1 -14     1 3                 2 -16   1 2                   -1 -18   1                     1 -20 1                       -1

Integral Khovanov Homology (db, data source)

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

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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