L11a107

L11a106

L11a108

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a107 at Knotilus!

Link Presentations

[edit Notes on L11a107's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{7}-2t(1)t(2)^{6}+2t(2)^{6}+2t(1)t(2)^{5}-2t(2)^{5}-2t(1)t(2)^{4}+2t(2)^{4}+2t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-2t(2)-t(1)}{{\sqrt {t(1)}}t(2)^{7/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {2}{q^{27/2}}}+{\frac {4}{q^{25/2}}}-{\frac {5}{q^{23/2}}}+{\frac {7}{q^{21/2}}}-{\frac {8}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {4}{q^{13/2}}}-{\frac {4}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-4za^{13}-3a^{13}z^{-1}+3z^{5}a^{11}+15z^{3}a^{11}+20za^{11}+7a^{11}z^{-1}-2z^{7}a^{9}-12z^{5}a^{9}-23z^{3}a^{9}-17za^{9}-4a^{9}z^{-1}-z^{7}a^{7}-5z^{5}a^{7}-6z^{3}a^{7}-za^{7}$ (db)
Kauffman polynomial	$a^{18}z^{4}-2a^{18}z^{2}+a^{18}+2a^{17}z^{5}-3a^{17}z^{3}+2a^{16}z^{6}-a^{16}z^{4}-2a^{16}z^{2}+2a^{15}z^{7}-2a^{15}z^{5}+a^{15}z^{3}+2a^{14}z^{8}-4a^{14}z^{6}+4a^{14}z^{4}+2a^{13}z^{9}-7a^{13}z^{7}+12a^{13}z^{5}-12a^{13}z^{3}+10a^{13}z-3a^{13}z^{-1}+a^{12}z^{10}-a^{12}z^{8}-9a^{12}z^{6}+24a^{12}z^{4}-22a^{12}z^{2}+7a^{12}+5a^{11}z^{9}-27a^{11}z^{7}+55a^{11}z^{5}-57a^{11}z^{3}+29a^{11}z-7a^{11}z^{-1}+a^{10}z^{10}-a^{10}z^{8}-12a^{10}z^{6}+27a^{10}z^{4}-23a^{10}z^{2}+7a^{10}+3a^{9}z^{9}-17a^{9}z^{7}+34a^{9}z^{5}-35a^{9}z^{3}+18a^{9}z-4a^{9}z^{-1}+2a^{8}z^{8}-9a^{8}z^{6}+9a^{8}z^{4}-a^{8}z^{2}+a^{7}z^{7}-5a^{7}z^{5}+6a^{7}z^{3}-a^{7}z$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                     2 1 -1 -10                   2     2 -12                 2 2     0 -14               5 2       3 -16             3 3         0 -18           5 4           1 -20         2 3             1 -22       3 5               -2 -24     1 2                 1 -26   1 3                   -2 -28   1                     1 -30 1                       -1

Integral Khovanov Homology (db, data source)

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

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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