L11a135

L11a134

L11a136

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a135 at Knotilus!

Link Presentations

[edit Notes on L11a135's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X14,3,15,4 X10,5,11,6 X20,11,21,12 X22,13,5,14 X12,21,13,22 X4,19,1,20 X18,16,19,15 X16,8,17,7 X8,18,9,17
Gauss code	{1, -2, 3, -8}, {4, -1, 10, -11, 2, -4, 5, -7, 6, -3, 9, -10, 11, -9, 8, -5, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{5}-2t(2)^{5}-6t(1)t(2)^{4}+6t(2)^{4}+10t(1)t(2)^{3}-9t(2)^{3}-9t(1)t(2)^{2}+10t(2)^{2}+6t(1)t(2)-6t(2)-2t(1)+2}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {19}{q^{5/2}}}-{\frac {23}{q^{7/2}}}+{\frac {22}{q^{9/2}}}-{\frac {20}{q^{11/2}}}+{\frac {15}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {4}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+2a^{7}z^{3}+a^{7}z-a^{5}z^{7}-3a^{5}z^{5}-3a^{5}z^{3}-a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z-a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+z^{3}a^{11}-4z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-8z^{7}a^{9}+12z^{5}a^{9}-7z^{3}a^{9}+za^{9}-9z^{8}a^{8}+10z^{6}a^{8}-z^{4}a^{8}-z^{2}a^{8}-6z^{9}a^{7}-2z^{7}a^{7}+14z^{5}a^{7}-6z^{3}a^{7}-2z^{10}a^{6}-13z^{8}a^{6}+27z^{6}a^{6}-14z^{4}a^{6}+3z^{2}a^{6}-12z^{9}a^{5}+19z^{7}a^{5}-9z^{5}a^{5}+4z^{3}a^{5}+za^{5}-a^{5}z^{-1}-2z^{10}a^{4}-11z^{8}a^{4}+32z^{6}a^{4}-24z^{4}a^{4}+4z^{2}a^{4}+a^{4}-6z^{9}a^{3}+9z^{7}a^{3}-4z^{3}a^{3}+2za^{3}-a^{3}z^{-1}-7z^{8}a^{2}+18z^{6}a^{2}-14z^{4}a^{2}+2z^{2}a^{2}-4z^{7}a+10z^{5}a-6z^{3}a-z^{6}+2z^{4}$ (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                   5 1   -4 -2                 9 3     6 -4               11 6       -5 -6             12 8         4 -8           10 11           1 -10         10 12             -2 -12       6 11               5 -14     3 9                 -6 -16   1 6                   5 -18   3                     -3 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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