L11a40

L11a39

L11a41

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a40 at Knotilus!

Link Presentations

[edit Notes on L11a40's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{5}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-8q^{9/2}+14q^{7/2}-19q^{5/2}+20q^{3/2}-21{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+6az^{3}-9z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-2a^{3}z+8az-10za^{-1}+5za^{-3}-za^{-5}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-6z^{4}a^{-6}+3z^{2}a^{-6}+7z^{7}a^{-5}-10z^{5}a^{-5}+6z^{3}a^{-5}-2za^{-5}+7z^{8}a^{-4}+a^{4}z^{6}-3z^{6}a^{-4}-3a^{4}z^{4}-10z^{4}a^{-4}+3a^{4}z^{2}+9z^{2}a^{-4}-a^{4}-a^{-4}+4z^{9}a^{-3}+3a^{3}z^{7}+10z^{7}a^{-3}-8a^{3}z^{5}-31z^{5}a^{-3}+8a^{3}z^{3}+26z^{3}a^{-3}-4a^{3}z-10za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+z^{10}a^{-2}+4a^{2}z^{8}+14z^{8}a^{-2}-5a^{2}z^{6}-21z^{6}a^{-2}-5a^{2}z^{4}-3z^{4}a^{-2}+10a^{2}z^{2}+13z^{2}a^{-2}-4a^{2}-4a^{-2}+3az^{9}+7z^{9}a^{-1}+5az^{7}+5z^{7}a^{-1}-26az^{5}-38z^{5}a^{-1}+29az^{3}+40z^{3}a^{-1}-15az-19za^{-1}+4az^{-1}+4a^{-1}z^{-1}+z^{10}+11z^{8}-20z^{6}-z^{4}+14z^{2}-7$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 9 3     -6 6               10 5       5 4             10 9         -1 2           11 10           1 0         8 12             4 -2       5 9               -4 -4     2 8                 6 -6   1 5                   -4 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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