L11a134

L11a133

L11a135

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a134 at Knotilus!

Link Presentations

[edit Notes on L11a134's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+4t(1)t(2)^{4}-6t(2)^{4}-10t(1)t(2)^{3}+12t(2)^{3}+12t(1)t(2)^{2}-10t(2)^{2}-6t(1)t(2)+4t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+9{\sqrt {q}}-{\frac {14}{\sqrt {q}}}+{\frac {18}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {20}{q^{7/2}}}-{\frac {18}{q^{9/2}}}+{\frac {13}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}-a^{7}z^{-1}-z^{5}a^{5}+z^{3}a^{5}+5za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-8z^{3}a^{3}-10za^{3}-4a^{3}z^{-1}-z^{5}a+z^{3}a+3za+az^{-1}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-3a^{9}z^{5}+3a^{9}z^{3}-a^{9}z+4a^{8}z^{8}-14a^{8}z^{6}+15a^{8}z^{4}-4a^{8}z^{2}-a^{8}+5a^{7}z^{9}-13a^{7}z^{7}+4a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z+a^{7}z^{-1}+2a^{6}z^{10}+8a^{6}z^{8}-39a^{6}z^{6}+35a^{6}z^{4}-2a^{6}z^{2}-4a^{6}+12a^{5}z^{9}-23a^{5}z^{7}-8a^{5}z^{5}+27a^{5}z^{3}-14a^{5}z+4a^{5}z^{-1}+2a^{4}z^{10}+16a^{4}z^{8}-47a^{4}z^{6}+24a^{4}z^{4}+8a^{4}z^{2}-7a^{4}+7a^{3}z^{9}+4a^{3}z^{7}-37a^{3}z^{5}+34a^{3}z^{3}-15a^{3}z+4a^{3}z^{-1}+12a^{2}z^{8}-13a^{2}z^{6}-5a^{2}z^{4}+z^{4}a^{-2}+9a^{2}z^{2}-4a^{2}+13az^{7}-18az^{5}+4z^{5}a^{-1}+11az^{3}-z^{3}a^{-1}-4az+az^{-1}+9z^{6}-8z^{4}+3z^{2}-1$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   6 1   -5 0                 8 3     5 -2               11 7       -4 -4             11 7         4 -6           9 11           2 -8         9 11             -2 -10       5 10               5 -12     3 8                 -5 -14   1 5                   4 -16   3                     -3 -18 1                       1

Integral Khovanov Homology (db, data source)

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

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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