L10a171

L10a170

L10a172

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a171 at Knotilus!

Link Presentations

[edit Notes on L10a171's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{2}x-uvw^{2}+uvwx^{2}-3uvwx+2uvw+uvx-uv-uw^{2}x-uwx^{2}+3uwx-uw+ux^{2}-2ux+u+vw^{2}x^{2}-2vw^{2}x+vw^{2}-vwx^{2}+3vwx-vw-vx-w^{2}x^{2}+w^{2}x+2wx^{2}-3wx+w-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {14}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}+a^{9}z^{-3}-3z^{3}a^{7}-6za^{7}-6a^{7}z^{-1}-3a^{7}z^{-3}+2z^{5}a^{5}+6z^{3}a^{5}+10za^{5}+9a^{5}z^{-1}+3a^{5}z^{-3}+z^{5}a^{3}-4za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-z^{2}a^{10}-5z^{7}a^{9}+10z^{5}a^{9}-10z^{3}a^{9}+11za^{9}-5a^{9}z^{-1}+a^{9}z^{-3}-4z^{8}a^{8}+2z^{6}a^{8}+9z^{4}a^{8}-14z^{2}a^{8}-3a^{8}z^{-2}+10a^{8}-z^{9}a^{7}-13z^{7}a^{7}+42z^{5}a^{7}-54z^{3}a^{7}+33za^{7}-12a^{7}z^{-1}+3a^{7}z^{-3}-8z^{8}a^{6}+10z^{6}a^{6}+8z^{4}a^{6}-26z^{2}a^{6}-6a^{6}z^{-2}+19a^{6}-z^{9}a^{5}-13z^{7}a^{5}+42z^{5}a^{5}-54z^{3}a^{5}+33za^{5}-12a^{5}z^{-1}+3a^{5}z^{-3}-4z^{8}a^{4}+2z^{6}a^{4}+9z^{4}a^{4}-14z^{2}a^{4}-3a^{4}z^{-2}+10a^{4}-5z^{7}a^{3}+10z^{5}a^{3}-10z^{3}a^{3}+11za^{3}-5a^{3}z^{-1}+a^{3}z^{-3}-3z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                     1 1 0                   2   -2 -2                 5 1   4 -4               6 4     -2 -6             8 3       5 -8           6 8         2 -10         8 6           2 -12       3 8             5 -14     4 6               -2 -16   1 5                 4 -18   2                   -2 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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