L11a106

L11a105

L11a107

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a106 at Knotilus!

Link Presentations

[edit Notes on L11a106's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{5}-4uv^{4}+8uv^{3}-8uv^{2}+6uv-2u-2v^{5}+6v^{4}-8v^{3}+8v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {16}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {18}{q^{9/2}}}-{\frac {16}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-6a^{7}z-3a^{7}z^{-1}+3a^{5}z^{5}+9a^{5}z^{3}+9a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-7a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-3z^{2}a^{10}+a^{10}-5z^{7}a^{9}+7z^{5}a^{9}-4z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-5z^{8}a^{8}+2z^{6}a^{8}+7z^{4}a^{8}-8z^{2}a^{8}+3a^{8}-3z^{9}a^{7}-7z^{7}a^{7}+22z^{5}a^{7}-22z^{3}a^{7}+12za^{7}-3a^{7}z^{-1}-z^{10}a^{6}-10z^{8}a^{6}+18z^{6}a^{6}-6z^{4}a^{6}-5z^{2}a^{6}+3a^{6}-7z^{9}a^{5}+4z^{7}a^{5}+17z^{5}a^{5}-26z^{3}a^{5}+17za^{5}-4a^{5}z^{-1}-z^{10}a^{4}-11z^{8}a^{4}+30z^{6}a^{4}-22z^{4}a^{4}+2z^{2}a^{4}+2a^{4}-4z^{9}a^{3}+2z^{7}a^{3}+14z^{5}a^{3}-17z^{3}a^{3}+8za^{3}-2a^{3}z^{-1}-6z^{8}a^{2}+16z^{6}a^{2}-12z^{4}a^{2}+2z^{2}a^{2}-4z^{7}a+11z^{5}a-7z^{3}a-z^{6}+2z^{4}$ (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                   4 1   -3 -2                 8 3     5 -4               9 5       -4 -6             10 7         3 -8           8 9           1 -10         8 10             -2 -12       5 9               4 -14     2 7                 -5 -16   1 5                   4 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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