L11a133

L11a132

L11a134

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a133 at Knotilus!

Link Presentations

[edit Notes on L11a133's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}+4t(1)t(2)^{4}-6t(2)^{4}-14t(1)t(2)^{3}+14t(2)^{3}+14t(1)t(2)^{2}-14t(2)^{2}-6t(1)t(2)+4t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {25}{q^{9/2}}}-{\frac {22}{q^{7/2}}}+{\frac {15}{q^{5/2}}}-{\frac {10}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {5}{q^{19/2}}}+{\frac {10}{q^{17/2}}}-{\frac {16}{q^{15/2}}}+{\frac {22}{q^{13/2}}}-{\frac {25}{q^{11/2}}}-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}+za^{9}+a^{9}z^{-1}+z^{5}a^{7}-2z^{3}a^{7}-5za^{7}-3a^{7}z^{-1}+3z^{5}a^{5}+6z^{3}a^{5}+7za^{5}+4a^{5}z^{-1}+z^{5}a^{3}-2z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}-z^{3}a$ (db)
Kauffman polynomial	$-z^{6}a^{12}+z^{4}a^{12}-5z^{7}a^{11}+10z^{5}a^{11}-4z^{3}a^{11}-za^{11}-9z^{8}a^{10}+20z^{6}a^{10}-12z^{4}a^{10}+z^{2}a^{10}+a^{10}-7z^{9}a^{9}+4z^{7}a^{9}+17z^{5}a^{9}-14z^{3}a^{9}+3za^{9}-a^{9}z^{-1}-2z^{10}a^{8}-19z^{8}a^{8}+52z^{6}a^{8}-36z^{4}a^{8}+5z^{2}a^{8}+3a^{8}-14z^{9}a^{7}+13z^{7}a^{7}+24z^{5}a^{7}-33z^{3}a^{7}+13za^{7}-3a^{7}z^{-1}-2z^{10}a^{6}-21z^{8}a^{6}+52z^{6}a^{6}-38z^{4}a^{6}+5z^{2}a^{6}+3a^{6}-7z^{9}a^{5}-5z^{7}a^{5}+33z^{5}a^{5}-36z^{3}a^{5}+16za^{5}-4a^{5}z^{-1}-11z^{8}a^{4}+17z^{6}a^{4}-11z^{4}a^{4}+z^{2}a^{4}+2a^{4}-9z^{7}a^{3}+15z^{5}a^{3}-12z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}-4z^{6}a^{2}+4z^{4}a^{2}-z^{5}a+z^{3}a$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   7 1   6 -4                 9 4     -5 -6               13 6       7 -8             12 9         -3 -10           13 13           0 -12         10 13             3 -14       6 12               -6 -16     4 10                 6 -18   1 6                   -5 -20   4                     4 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-6$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-4$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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