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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- 2q-15/2 + 4q-13/2 - 6q-11/2 + 8q-9/2 - 9q-7/2 + 8q-5/2 - 7q-3/2 + 4q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 + q-26 + q-24 + q-22 - 2q-20 - 2q-16 + 2q-12 + 4q-8 + 2q-4 + q-2 + q2 - q4

HOMFLY-PT Polynomial:

az + 3az3 + az5 - 2a3z-1 - 8a3z - 9a3z3 - 5a3z5 - a3z7 + 4a5z-1 + 9a5z + 8a5z3 + 2a5z5 - 3a7z-1 - 4a7z - a7z3 + a9z-1

Kauffman Polynomial:

- z2 + 3z4 - z6 + 3az - 10az3 + 11az5 - 3az7 - a2 + a2z2 - 5a2z4 + 9a2z6 - 3a2z8 - 2a3z-1 + 16a3z - 33a3z3 + 25a3z5 - 4a3z7 - a3z9 - 2a4 + 9a4z2 - 22a4z4 + 20a4z6 - 6a4z8 - 4a5z-1 + 24a5z - 39a5z3 + 23a5z5 - 4a5z7 - a5z9 - 3a6 + 11a6z2 - 16a6z4 + 9a6z6 - 3a6z8 - 3a7z-1 + 15a7z - 19a7z3 + 9a7z5 - 3a7z7 - a8 + 4a8z2 - 2a8z4 - a8z6 - a9z-1 + 4a9z - 3a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                   1 j = 2                 2   j = 0               2 1   j = -2             5 2     j = -4           4 3       j = -6         5 4         j = -8       3 4           j = -10     3 5             j = -12   2 4               j = -14   2                 j = -16 2

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, 4]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 4]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 14, 10, 15], > X[3, 8, 4, 9], X[5, 11, 6, 10], X[11, 19, 12, 18], X[13, 20, 14, 21], > X[19, 5, 20, 22], X[21, 12, 22, 13], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, -11, -2, 3, 7, > -9, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 6 8 9 8 7 4 3/2 ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + q 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 -22 2 2 2 4 2 -2 2 4 q + q + q + q - --- - --- + --- + -- + -- + q + q - q 20 16 12 8 4 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 4]][a, z]
Out[8]=	3 5 7 9 -2 a 4 a 3 a a 3 5 7 3 3 3 ----- + ---- - ---- + -- + a z - 8 a z + 9 a z - 4 a z + 3 a z - 9 a z + z z z z 5 3 7 3 5 3 5 5 5 3 7 > 8 a z - a z + a z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 4]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 2 a 4 a 3 a a 3 5 -a - 2 a - 3 a - a - ---- - ---- - ---- - -- + 3 a z + 16 a z + 24 a z + z z z z 7 9 2 2 2 4 2 6 2 8 2 3 > 15 a z + 4 a z - z + a z + 9 a z + 11 a z + 4 a z - 10 a z - 3 3 5 3 7 3 9 3 4 2 4 4 4 > 33 a z - 39 a z - 19 a z - 3 a z + 3 z - 5 a z - 22 a z - 6 4 8 4 5 3 5 5 5 7 5 6 > 16 a z - 2 a z + 11 a z + 25 a z + 23 a z + 9 a z - z + 2 6 4 6 6 6 8 6 7 3 7 5 7 > 9 a z + 20 a z + 9 a z - a z - 3 a z - 4 a z - 4 a z - 7 7 2 8 4 8 6 8 3 9 5 9 > 3 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 2 2 2 4 3 5 3 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 5 4 4 2 t 2 2 2 4 3 > ----- + ---- + ---- + 2 t + --- + t + 2 q t + q t 6 2 6 4 2 q t q t q t q

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