L11a171

L11a170

L11a172

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a171 at Knotilus!

Link Presentations

[edit Notes on L11a171's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-7t(1)^{2}t(2)^{3}+12t(1)t(2)^{3}-5t(2)^{3}+9t(1)^{2}t(2)^{2}-17t(1)t(2)^{2}+9t(2)^{2}-5t(1)^{2}t(2)+12t(1)t(2)-7t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-5{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {20}{q^{3/2}}}+{\frac {26}{q^{5/2}}}-{\frac {31}{q^{7/2}}}+{\frac {31}{q^{9/2}}}-{\frac {27}{q^{11/2}}}+{\frac {20}{q^{13/2}}}-{\frac {12}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+a^{7}z^{3}-a^{5}z^{7}-2a^{5}z^{5}-a^{5}z^{3}-a^{3}z^{7}-2a^{3}z^{5}-a^{3}z^{3}+a^{3}z^{-1}+az^{5}+az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}+5a^{10}z^{6}-3a^{10}z^{4}+12a^{9}z^{7}-14a^{9}z^{5}+5a^{9}z^{3}+17a^{8}z^{8}-25a^{8}z^{6}+13a^{8}z^{4}-2a^{8}z^{2}+13a^{7}z^{9}-7a^{7}z^{7}-14a^{7}z^{5}+8a^{7}z^{3}+4a^{6}z^{10}+26a^{6}z^{8}-64a^{6}z^{6}+38a^{6}z^{4}-6a^{6}z^{2}+23a^{5}z^{9}-34a^{5}z^{7}+a^{5}z^{5}+6a^{5}z^{3}+4a^{4}z^{10}+19a^{4}z^{8}-54a^{4}z^{6}+34a^{4}z^{4}-6a^{4}z^{2}+10a^{3}z^{9}-10a^{3}z^{7}-9a^{3}z^{5}+8a^{3}z^{3}+a^{3}z-a^{3}z^{-1}+10a^{2}z^{8}-19a^{2}z^{6}+11a^{2}z^{4}-2a^{2}z^{2}+a^{2}+5az^{7}-9az^{5}+5az^{3}+az-az^{-1}+z^{6}-z^{4}$ (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 3 χ 4                       1 -1 2                     4   4 0                   7 1   -6 -2                 13 4     9 -4               14 8       -6 -6             17 12         5 -8           15 15           0 -10         12 16             -4 -12       8 15               7 -14     4 12                 -8 -16   1 8                   7 -18   4                     -4 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-4$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-3$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=-2$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{17}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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