L10a115

L10a114

L10a116

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a115 at Knotilus!

Link Presentations

[edit Notes on L10a115's Link Presentations]

Planar diagram presentation	X12,1,13,2 X20,9,11,10 X14,3,15,4 X16,5,17,6 X4,15,5,16 X8,17,9,18 X18,7,19,8 X6,19,7,20 X2,11,3,12 X10,13,1,14
Gauss code	{1, -9, 3, -5, 4, -8, 7, -6, 2, -10}, {9, -1, 10, -3, 5, -4, 6, -7, 8, -2}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}\left(-v^{2}\right)-2u^{3}v^{3}+2u^{3}v^{2}-u^{3}v-u^{2}v^{4}+2u^{2}v^{3}-3u^{2}v^{2}+2u^{2}v-u^{2}-uv^{3}+2uv^{2}-2uv-v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{23/2}}}-{\frac {3}{q^{21/2}}}+{\frac {5}{q^{19/2}}}-{\frac {6}{q^{17/2}}}+{\frac {7}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {5}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}+3za^{11}+a^{11}z^{-1}-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}-a^{9}z^{-1}-2z^{5}a^{7}-7z^{3}a^{7}-5za^{7}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+4z^{3}a^{15}-4za^{15}-z^{6}a^{14}+2z^{4}a^{14}-z^{7}a^{13}+z^{5}a^{13}+z^{3}a^{13}-z^{8}a^{12}+2z^{6}a^{12}-5z^{4}a^{12}+6z^{2}a^{12}-z^{9}a^{11}+4z^{7}a^{11}-12z^{5}a^{11}+13z^{3}a^{11}-6za^{11}+a^{11}z^{-1}-3z^{8}a^{10}+10z^{6}a^{10}-16z^{4}a^{10}+7z^{2}a^{10}-a^{10}-z^{9}a^{9}+2z^{7}a^{9}-3z^{5}a^{9}+2z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-2z^{8}a^{8}+5z^{6}a^{8}-4z^{4}a^{8}-3z^{7}a^{7}+10z^{5}a^{7}-11z^{3}a^{7}+5za^{7}-2z^{6}a^{6}+5z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-za^{5}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   2 1 -1 -8                 2     2 -10               3 2     -1 -12             4 2       2 -14           3 3         0 -16         3 4           -1 -18       2 3             1 -20     1 3               -2 -22     2                 2 -24 1 1                   0 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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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