L11a276

L11a275

L11a277

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a276 at Knotilus!

Link Presentations

[edit Notes on L11a276's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X18,7,19,8 X22,20,9,19 X20,16,21,15 X16,22,17,21 X8,9,1,10 X4,13,5,14 X6,17,7,18
Gauss code	{1, -2, 3, -10, 4, -11, 5, -9}, {9, -1, 2, -3, 10, -4, 7, -8, 11, -5, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{3}t(2)^{5}-2t(1)^{3}t(2)^{4}+2t(1)^{2}t(2)^{4}+2t(1)^{3}t(2)^{3}-3t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+2t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-2t(2)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {2}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {5}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{5}z^{9}+a^{7}z^{7}-8a^{5}z^{7}+a^{3}z^{7}+6a^{7}z^{5}-24a^{5}z^{5}+6a^{3}z^{5}+12a^{7}z^{3}-34a^{5}z^{3}+11a^{3}z^{3}+10a^{7}z-21a^{5}z+7a^{3}z+2a^{7}z^{-1}-3a^{5}z^{-1}+a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}+za^{13}-2z^{4}a^{12}+z^{2}a^{12}-3z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-3z^{6}a^{10}+3z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+5z^{5}a^{9}-2z^{3}a^{9}+za^{9}-3z^{8}a^{8}+8z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-3z^{9}a^{7}+13z^{7}a^{7}-22z^{5}a^{7}+24z^{3}a^{7}-12za^{7}+2a^{7}z^{-1}-z^{10}a^{6}+16z^{6}a^{6}-30z^{4}a^{6}+19z^{2}a^{6}-3a^{6}-5z^{9}a^{5}+28z^{7}a^{5}-53z^{5}a^{5}+48z^{3}a^{5}-23za^{5}+3a^{5}z^{-1}-z^{10}a^{4}+2z^{8}a^{4}+11z^{6}a^{4}-30z^{4}a^{4}+20z^{2}a^{4}-3a^{4}-2z^{9}a^{3}+12z^{7}a^{3}-23z^{5}a^{3}+18z^{3}a^{3}-7za^{3}+a^{3}z^{-1}-z^{8}a^{2}+6z^{6}a^{2}-11z^{4}a^{2}+7z^{2}a^{2}-a^{2}$ (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 2                       1 -1 0                     1   1 -2                   2 1   -1 -4                 3 1     2 -6               3 3       0 -8             4 2         2 -10           3 3           0 -12         3 4             -1 -14       2 3               1 -16     2 3                 -1 -18     2                   2 -20 1 2                     -1 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a275 L11a277