L11a77

L11a76

L11a78

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a77 at Knotilus!

Link Presentations

[edit Notes on L11a77's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-3v+1\right)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+8q^{5/2}-13q^{3/2}+16{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {19}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}+z^{3}a^{-3}-7a^{3}z+za^{-3}-3a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+8az^{3}-5z^{3}a^{-1}+8az-4za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-7az^{9}-4z^{9}a^{-1}-5a^{4}z^{8}-11a^{2}z^{8}-6z^{8}a^{-2}-12z^{8}-5a^{5}z^{7}-8a^{3}z^{7}-z^{7}a^{-1}-4z^{7}a^{-3}-3a^{6}z^{6}+2a^{4}z^{6}+20a^{2}z^{6}+13z^{6}a^{-2}-z^{6}a^{-4}+29z^{6}-a^{7}z^{5}+7a^{5}z^{5}+25a^{3}z^{5}+29az^{5}+22z^{5}a^{-1}+10z^{5}a^{-3}+5a^{6}z^{4}+6a^{4}z^{4}-7a^{2}z^{4}-5z^{4}a^{-2}+2z^{4}a^{-4}-15z^{4}+2a^{7}z^{3}-4a^{5}z^{3}-26a^{3}z^{3}-36az^{3}-23z^{3}a^{-1}-7z^{3}a^{-3}-3a^{6}z^{2}-7a^{4}z^{2}-5a^{2}z^{2}-z^{2}a^{-2}-z^{2}a^{-4}-z^{2}-a^{7}z+2a^{5}z+13a^{3}z+17az+9za^{-1}+2za^{-3}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   5 1   -4 4                 8 3     5 2               8 5       -3 0             12 8         4 -2           9 10           1 -4         7 10             -3 -6       5 9               4 -8     2 7                 -5 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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