L11a76

L11a75

L11a77

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a76 at Knotilus!

Link Presentations

[edit Notes on L11a76's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4t(2)^{3}+6t(1)t(2)^{2}-9t(2)^{2}-9t(1)t(2)+6t(2)+4t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {11}{q^{13/2}}}-{\frac {12}{q^{15/2}}}+{\frac {9}{q^{17/2}}}-{\frac {7}{q^{19/2}}}+{\frac {5}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+3za^{11}+a^{11}z^{-1}-2z^{3}a^{9}+za^{9}+2a^{9}z^{-1}-4z^{3}a^{7}-4za^{7}-2a^{7}z^{-1}-3z^{3}a^{5}-2za^{5}-z^{3}a^{3}$ (db)
Kauffman polynomial	$a^{14}z^{8}-6a^{14}z^{6}+13a^{14}z^{4}-12a^{14}z^{2}+4a^{14}+2a^{13}z^{9}-10a^{13}z^{7}+15a^{13}z^{5}-6a^{13}z^{3}-a^{13}z^{-1}+a^{12}z^{10}+a^{12}z^{8}-23a^{12}z^{6}+48a^{12}z^{4}-34a^{12}z^{2}+9a^{12}+6a^{11}z^{9}-23a^{11}z^{7}+20a^{11}z^{5}+3a^{11}z^{3}-3a^{11}z-a^{11}z^{-1}+a^{10}z^{10}+7a^{10}z^{8}-38a^{10}z^{6}+46a^{10}z^{4}-19a^{10}z^{2}+4a^{10}+4a^{9}z^{9}-4a^{9}z^{7}-19a^{9}z^{5}+26a^{9}z^{3}-12a^{9}z+2a^{9}z^{-1}+7a^{8}z^{8}-13a^{8}z^{6}-2a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+9a^{7}z^{7}-18a^{7}z^{5}+11a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+8a^{6}z^{6}-10a^{6}z^{4}+2a^{6}z^{2}+6a^{5}z^{5}-5a^{5}z^{3}+2a^{5}z+3a^{4}z^{4}+a^{3}z^{3}$ (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 -2                       1 1 -4                     3 1 -2 -6                   3     3 -8                 5 3     -2 -10               6 3       3 -12             6 6         0 -14           6 5           1 -16         3 6             3 -18       4 6               -2 -20     1 3                 2 -22   1 4                   -3 -24   1                     1 -26 1                       -1

Integral Khovanov Homology (db, data source)

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

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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