L11a177

L11a176

L11a178

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a177 at Knotilus!

Link Presentations

[edit Notes on L11a177's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X16,7,17,8 X18,10,19,9 X20,11,21,12 X22,15,7,16 X12,21,13,22 X14,6,15,5 X4,14,5,13 X6,18,1,17 X2,19,3,20
Gauss code	{1, -11, 2, -9, 8, -10}, {3, -1, 4, -2, 5, -7, 9, -8, 6, -3, 10, -4, 11, -5, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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