L11a97

L11a96

L11a98

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a97 at Knotilus!

Link Presentations

[edit Notes on L11a97's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+6t(1)t(2)^{4}-6t(2)^{4}-13t(1)t(2)^{3}+13t(2)^{3}+13t(1)t(2)^{2}-13t(2)^{2}-6t(1)t(2)+6t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {16}{q^{11/2}}}-{\frac {23}{q^{13/2}}}+{\frac {25}{q^{15/2}}}-{\frac {25}{q^{17/2}}}+{\frac {22}{q^{19/2}}}-{\frac {15}{q^{21/2}}}+{\frac {9}{q^{23/2}}}-{\frac {4}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}+3z^{3}a^{11}+2za^{11}-2a^{11}z^{-1}-2z^{5}a^{9}+7za^{9}+5a^{9}z^{-1}-4z^{5}a^{7}-11z^{3}a^{7}-10za^{7}-3a^{7}z^{-1}-z^{5}a^{5}-z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+2z^{4}a^{16}-z^{2}a^{16}-4z^{7}a^{15}+9z^{5}a^{15}-6z^{3}a^{15}+za^{15}-7z^{8}a^{14}+15z^{6}a^{14}-11z^{4}a^{14}+4z^{2}a^{14}-a^{14}-6z^{9}a^{13}+3z^{7}a^{13}+14z^{5}a^{13}-13z^{3}a^{13}+2za^{13}-2z^{10}a^{12}-17z^{8}a^{12}+48z^{6}a^{12}-41z^{4}a^{12}+14z^{2}a^{12}-14z^{9}a^{11}+15z^{7}a^{11}+12z^{5}a^{11}-14z^{3}a^{11}+4za^{11}-2a^{11}z^{-1}-2z^{10}a^{10}-22z^{8}a^{10}+52z^{6}a^{10}-34z^{4}a^{10}+2z^{2}a^{10}+5a^{10}-8z^{9}a^{9}-2z^{7}a^{9}+25z^{5}a^{9}-24z^{3}a^{9}+13za^{9}-5a^{9}z^{-1}-12z^{8}a^{8}+16z^{6}a^{8}-3z^{4}a^{8}-7z^{2}a^{8}+5a^{8}-10z^{7}a^{7}+17z^{5}a^{7}-16z^{3}a^{7}+10za^{7}-3a^{7}z^{-1}-4z^{6}a^{6}+3z^{4}a^{6}-z^{5}a^{5}+z^{3}a^{5}$ (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 -4                       1 1 -6                     4 1 -3 -8                   7     7 -10                 9 4     -5 -12               14 7       7 -14             12 10         -2 -16           13 13           0 -18         9 12             3 -20       6 13               -7 -22     3 9                 6 -24   1 6                   -5 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-8$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-7$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-6$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-5$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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