L10a111

L10a110

L10a112

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a111 at Knotilus!

Link Presentations

[edit Notes on L10a111's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,7,17,8 X8394 X2,18,3,17 X14,6,15,5 X6,12,7,11 X18,10,19,9 X20,15,11,16 X10,13,1,14 X4,19,5,20
Gauss code	{1, -4, 3, -10, 5, -6, 2, -3, 7, -9}, {6, -1, 9, -5, 8, -2, 4, -7, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}-q^{7/2}+{\frac {15}{q^{7/2}}}+5q^{5/2}-{\frac {19}{q^{5/2}}}-11q^{3/2}+{\frac {20}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-z^{5}a^{-1}+a^{5}z^{3}-4a^{3}z^{3}+3az^{3}-z^{3}a^{-1}+a^{5}z-2a^{3}z+az+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-4a^{3}z^{9}-4az^{9}-9a^{4}z^{8}-20a^{2}z^{8}-11z^{8}-8a^{5}z^{7}-12a^{3}z^{7}-15az^{7}-11z^{7}a^{-1}-4a^{6}z^{6}+11a^{4}z^{6}+34a^{2}z^{6}-5z^{6}a^{-2}+14z^{6}-a^{7}z^{5}+12a^{5}z^{5}+34a^{3}z^{5}+39az^{5}+17z^{5}a^{-1}-z^{5}a^{-3}+5a^{6}z^{4}-3a^{4}z^{4}-14a^{2}z^{4}+4z^{4}a^{-2}-2z^{4}+a^{7}z^{3}-8a^{5}z^{3}-22a^{3}z^{3}-19az^{3}-6z^{3}a^{-1}-2a^{6}z^{2}-a^{4}z^{2}+2a^{2}z^{2}+z^{2}+2a^{5}z+4a^{3}z+2az+1-az^{-1}-a^{-1}z^{-1}$ (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 χ 8                     1 1 6                   4   -4 4                 7 1   6 2               8 4     -4 0             12 7       5 -2           10 10         0 -4         9 10           -1 -6       6 10             4 -8     3 9               -6 -10   1 6                 5 -12   3                   -3 -14 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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