L11a42

L11a41

L11a43

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a42 at Knotilus!

Link Presentations

[edit Notes on L11a42's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+4v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {7}{q^{9/2}}}-3q^{7/2}+{\frac {12}{q^{7/2}}}+6q^{5/2}-{\frac {16}{q^{5/2}}}-12q^{3/2}+{\frac {18}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+15{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-6a^{3}z^{3}+8az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}+2a^{5}z-7a^{3}z+9az-6za^{-1}+2za^{-3}+a^{5}z^{-1}-3a^{3}z^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-6az^{9}-3z^{9}a^{-1}-5a^{4}z^{8}-9a^{2}z^{8}-4z^{8}a^{-2}-8z^{8}-5a^{5}z^{7}-7a^{3}z^{7}-z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+2a^{4}z^{6}+14a^{2}z^{6}+8z^{6}a^{-2}-z^{6}a^{-4}+18z^{6}-a^{7}z^{5}+7a^{5}z^{5}+23a^{3}z^{5}+24az^{5}+18z^{5}a^{-1}+9z^{5}a^{-3}+5a^{6}z^{4}+7a^{4}z^{4}-2z^{4}a^{-2}+3z^{4}a^{-4}-7z^{4}+2a^{7}z^{3}-4a^{5}z^{3}-24a^{3}z^{3}-33az^{3}-24z^{3}a^{-1}-9z^{3}a^{-3}-3a^{6}z^{2}-8a^{4}z^{2}-8a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}a^{-4}-3z^{2}-a^{7}z+2a^{5}z+13a^{3}z+19az+13za^{-1}+4za^{-3}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 8 2     6 2               7 4       -3 0             11 8         3 -2           9 9           0 -4         7 9             -2 -6       5 9               4 -8     2 7                 -5 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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