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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X3849 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X15,2,16,3

Gauss Code:

{{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -7, 8, -11, -2, 3, 10, -8, 7, -9, 6}}

Jones Polynomial:

q-23/2 - q-21/2 + 2q-19/2 - 3q-17/2 + 3q-15/2 - 4q-13/2 + 2q-11/2 - 2q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-40 - q-38 - q-36 - q-34 + q-28 + q-26 + q-24 + 2q-22 + q-20 + 2q-18 + 2q-16 + q-14 + q-12 + q-8

HOMFLY-PT Polynomial:

- a5z-1 - 3a5z - 4a5z3 - a5z5 + a7z-1 - 3a7z3 - a7z5 - a9z-1 - 3a9z - 4a9z3 - a9z5 + 2a11z-1 + 4a11z + a11z3 - a13z-1

Kauffman Polynomial:

a5z-1 - 3a5z + 4a5z3 - a5z5 - a6 + 3a6z4 - a6z6 + a7z-1 - 3a7z + 3a7z5 - a7z7 - 4a8z4 + 4a8z6 - a8z8 + a9z-1 - 6a9z + 11a9z3 - 13a9z5 + 6a9z7 - a9z9 - 4a10 + 17a10z2 - 25a10z4 + 12a10z6 - 2a10z8 + 2a11z-1 - 8a11z + 18a11z3 - 18a11z5 + 7a11z7 - a11z9 - 7a12 + 21a12z2 - 19a12z4 + 7a12z6 - a12z8 + a13z-1 - 2a13z + 3a13z3 - a13z5 - 3a14 + 4a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   1 j = -6                 1 1 j = -8               1     j = -10           1 2 1     j = -12           3 1       j = -14         1 2         j = -16       2 2           j = -18       1             j = -20   1 2               j = -22                     j = -24 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 13]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 13]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[11, 22, 12, 5], X[13, 20, 14, 21], X[19, 14, 20, 15], > X[21, 12, 22, 13], X[9, 18, 10, 19], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -7, 8, -11, -2, 3, 10, > -8, 7, -9, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) -(21/2) 2 3 3 4 2 2 -(7/2) q - q + ----- - ----- + ----- - ----- + ----- - ---- + q - 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q -(5/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -38 -36 -34 -28 -26 -24 2 -20 2 2 -q - q - q - q + q + q + q + --- + q + --- + --- + 22 18 16 q q q -14 -12 -8 > q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 13]][a, z]
Out[8]=	5 7 9 11 13 a a a 2 a a 5 9 11 5 3 7 3 -(--) + -- - -- + ----- - --- - 3 a z - 3 a z + 4 a z - 4 a z - 3 a z - z z z z z 9 3 11 3 5 5 7 5 9 5 > 4 a z + a z - a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 13]][a, z]
Out[9]=	5 7 9 11 13 6 10 12 14 a a a 2 a a 5 7 -a - 4 a - 7 a - 3 a + -- + -- + -- + ----- + --- - 3 a z - 3 a z - z z z z z 9 11 13 10 2 12 2 14 2 5 3 > 6 a z - 8 a z - 2 a z + 17 a z + 21 a z + 4 a z + 4 a z + 9 3 11 3 13 3 6 4 8 4 10 4 > 11 a z + 18 a z + 3 a z + 3 a z - 4 a z - 25 a z - 12 4 14 4 5 5 7 5 9 5 11 5 13 5 > 19 a z - a z - a z + 3 a z - 13 a z - 18 a z - a z - 6 6 8 6 10 6 12 6 7 7 9 7 11 7 > a z + 4 a z + 12 a z + 7 a z - a z + 6 a z + 7 a z - 8 8 10 8 12 8 9 9 11 9 > a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 2 1 2 2 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 20 8 20 7 18 6 16 6 16 5 14 5 q t q t q t q t q t q t q t 2 3 1 1 2 1 1 1 > ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 14 4 12 4 10 4 12 3 10 3 10 2 8 2 6 q t q t q t q t q t q t q t q t

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