L11a174

L11a173

L11a175

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a174 at Knotilus!

Link Presentations

[edit Notes on L11a174's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{4}-4u^{2}v^{3}+4u^{2}v^{2}-2u^{2}v-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u-2v^{3}+4v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}-q^{7/2}+{\frac {14}{q^{7/2}}}+3q^{5/2}-{\frac {17}{q^{5/2}}}-7q^{3/2}+{\frac {17}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+11{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+4az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+5a^{3}z^{3}+6az^{3}-3z^{3}a^{-1}-2a^{5}z+4az-3za^{-1}+a^{5}z^{-1}-2a^{3}z^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}-a^{7}z+4a^{6}z^{8}-9a^{6}z^{6}+3a^{6}z^{4}+a^{6}z^{2}+4a^{5}z^{9}-9a^{5}z^{7}+9a^{5}z^{5}-8a^{5}z^{3}+a^{5}z^{-1}+2a^{4}z^{10}-4a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}+9a^{3}z^{9}-26a^{3}z^{7}+35a^{3}z^{5}+z^{5}a^{-3}-18a^{3}z^{3}-2z^{3}a^{-3}-2a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+3a^{2}z^{8}-14a^{2}z^{6}+3z^{6}a^{-2}+19a^{2}z^{4}-5z^{4}a^{-2}-7a^{2}z^{2}+a^{2}+5az^{9}-8az^{7}+6z^{7}a^{-1}+2az^{5}-14z^{5}a^{-1}+11az^{3}+12z^{3}a^{-1}-8az-5za^{-1}+2az^{-1}+a^{-1}z^{-1}+7z^{8}-17z^{6}+18z^{4}-6z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   5 1   4 2                 6 2     -4 0               9 5       4 -2             9 7         -2 -4           8 8           0 -6         7 10             3 -8       4 7               -3 -10     2 7                 5 -12   1 4                   -3 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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