L10a173

L10a172

L10a174

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a173 at Knotilus!

Link Presentations

[edit Notes on L10a173's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)t(4)^{2}t(3)^{2}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(1)t(4)t(3)^{2}+t(1)t(2)t(4)t(3)^{2}-2t(2)t(4)t(3)^{2}+2t(4)t(3)^{2}-t(3)^{2}-t(1)t(4)^{2}t(3)+t(1)t(2)t(4)^{2}t(3)-2t(2)t(4)^{2}t(3)+2t(4)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-t(2)t(3)+3t(1)t(4)t(3)-4t(1)t(2)t(4)t(3)+3t(2)t(4)t(3)-4t(4)t(3)+t(3)+t(1)t(4)^{2}-t(1)t(2)t(4)^{2}+t(2)t(4)^{2}-t(4)^{2}+t(1)-2t(1)t(4)+2t(1)t(2)t(4)-t(2)t(4)+t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	${\frac {15}{q^{9/2}}}-{\frac {17}{q^{7/2}}}+{\frac {11}{q^{5/2}}}-{\frac {9}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {16}{q^{11/2}}}-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-3a^{7}z^{-3}-5a^{7}z-6a^{7}z^{-1}+2a^{5}z^{5}+5a^{5}z^{3}+3a^{5}z^{-3}+9a^{5}z+9a^{5}z^{-1}+a^{3}z^{5}-a^{3}z^{3}-a^{3}z^{-3}-5a^{3}z-4a^{3}z^{-1}-az^{3}$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-12a^{9}z^{5}+14a^{9}z^{3}-a^{9}z^{-3}-12a^{9}z+5a^{9}z^{-1}+5a^{8}z^{8}-a^{8}z^{6}-13a^{8}z^{4}+17a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+2a^{7}z^{9}+12a^{7}z^{7}-36a^{7}z^{5}+42a^{7}z^{3}-3a^{7}z^{-3}-29a^{7}z+12a^{7}z^{-1}+12a^{6}z^{8}-17a^{6}z^{6}-4a^{6}z^{4}+23a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}+2a^{5}z^{9}+14a^{5}z^{7}-40a^{5}z^{5}+39a^{5}z^{3}-3a^{5}z^{-3}-23a^{5}z+12a^{5}z^{-1}+7a^{4}z^{8}-9a^{4}z^{6}+7a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+8a^{3}z^{7}-16a^{3}z^{5}+12a^{3}z^{3}-a^{3}z^{-3}-7a^{3}z+5a^{3}z^{-1}+4a^{2}z^{6}-5a^{2}z^{4}+az^{5}-az^{3}$ (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                   3   -3 -2                 6 1   5 -4               6 4     -2 -6             11 5       6 -8           8 10         2 -10         8 7           1 -12       5 10             5 -14     3 6               -3 -16     5                 5 -18 1 3                   -2 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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