L11a275

L11a274

L11a276

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a275 at Knotilus!

Link Presentations

[edit Notes on L11a275's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X18,8,19,7 X22,16,9,15 X20,18,21,17 X16,22,17,21 X8,20,1,19 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -5, 6, -10}, {5, -1, 2, -3, 11, -4, 7, -9, 8, -6, 10, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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