L11a501

L11a500

L11a502

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a501 at Knotilus!

Link Presentations

[edit Notes on L11a501's Link Presentations]

Planar diagram presentation	X8192 X14,5,15,6 X10,3,11,4 X4,13,5,14 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, {4, -2, 9, -11, 8, -7, 10, -9, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}vw^{3}-3u^{2}vw^{2}+3u^{2}vw-u^{2}v-u^{2}w^{3}+2u^{2}w^{2}-2u^{2}w+uv^{2}w^{3}-3uv^{2}w^{2}+3uv^{2}w-uv^{2}-2uvw^{3}+6uvw^{2}-6uvw+2uv+uw^{3}-3uw^{2}+3uw-u+2v^{2}w^{2}-2v^{2}w+v^{2}+vw^{3}-3vw^{2}+3vw-v}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+12q^{2}-16q+19-18q^{-1}+17q^{-2}-11q^{-3}+8q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+2a^{4}z^{2}-z^{2}a^{-4}+2a^{4}z^{-2}+3a^{4}-a^{-4}-a^{2}z^{6}-3a^{2}z^{4}+2z^{4}a^{-2}-7a^{2}z^{2}+3z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-9a^{2}+a^{-2}-z^{6}-z^{4}+2z^{2}+4z^{-2}+6$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+6a^{4}z^{8}+12a^{2}z^{8}+5z^{8}a^{-2}+11z^{8}+3a^{5}z^{7}-2a^{3}z^{7}-2az^{7}+8z^{7}a^{-1}+5z^{7}a^{-3}+a^{6}z^{6}-19a^{4}z^{6}-39a^{2}z^{6}-2z^{6}a^{-2}+3z^{6}a^{-4}-24z^{6}-7a^{5}z^{5}-17a^{3}z^{5}-29az^{5}-27z^{5}a^{-1}-7z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+26a^{4}z^{4}+51a^{2}z^{4}-6z^{4}a^{-2}-5z^{4}a^{-4}+21z^{4}+3a^{5}z^{3}+28a^{3}z^{3}+50az^{3}+31z^{3}a^{-1}+4z^{3}a^{-3}-2z^{3}a^{-5}+2a^{6}z^{2}-23a^{4}z^{2}-43a^{2}z^{2}+5z^{2}a^{-2}+3z^{2}a^{-4}-16z^{2}-19a^{3}z-35az-19za^{-1}-2za^{-3}+za^{-5}+11a^{4}+22a^{2}-a^{-4}+13+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     2   2 7                   5 1   -4 5                 7 2     5 3               9 5       -4 1             10 7         3 -1           10 11           1 -3         7 8             -1 -5       5 11               6 -7     3 6                 -3 -9     5                   5 -11 1 3                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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