L11a37

L11a36

L11a38

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a37 at Knotilus!

Link Presentations

[edit Notes on L11a37's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3t(2)^{3}+4t(1)t(2)^{2}-6t(2)^{2}-6t(1)t(2)+4t(2)+3t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {5}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {5}{q^{19/2}}}+{\frac {4}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+3za^{11}+2a^{11}z^{-1}-2z^{3}a^{9}-2za^{9}-a^{9}z^{-1}-2z^{3}a^{7}+a^{7}z^{-1}-2z^{3}a^{5}-2za^{5}-a^{5}z^{-1}-z^{3}a^{3}-za^{3}$ (db)
Kauffman polynomial	$-z^{8}a^{14}+6z^{6}a^{14}-12z^{4}a^{14}+10z^{2}a^{14}-3a^{14}-2z^{9}a^{13}+11z^{7}a^{13}-18z^{5}a^{13}+10z^{3}a^{13}-3za^{13}+a^{13}z^{-1}-z^{10}a^{12}+z^{8}a^{12}+16z^{6}a^{12}-39z^{4}a^{12}+28z^{2}a^{12}-7a^{12}-5z^{9}a^{11}+25z^{7}a^{11}-38z^{5}a^{11}+23z^{3}a^{11}-8za^{11}+2a^{11}z^{-1}-z^{10}a^{10}-z^{8}a^{10}+19z^{6}a^{10}-31z^{4}a^{10}+18z^{2}a^{10}-4a^{10}-3z^{9}a^{9}+11z^{7}a^{9}-14z^{5}a^{9}+13z^{3}a^{9}-5za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+6z^{6}a^{8}-3z^{7}a^{7}+3z^{5}a^{7}+4z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-3z^{6}a^{6}+2z^{4}a^{6}+z^{2}a^{6}-a^{6}-3z^{5}a^{5}+3z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-2z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+za^{3}$ (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                     2 1 -1 -6                   2     2 -8                 3 2     -1 -10               4 2       2 -12             4 4         0 -14           4 3           1 -16         2 4             2 -18       3 4               -1 -20     1 2                 1 -22   1 3                   -2 -24   1                     1 -26 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a36 L11a38