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DrawMorseLink

PD Presentation:

X8192 X3,10,4,11 X5,14,6,15 X16,8,17,7 X22,18,7,17 X15,13,16,12 X9,20,10,21 X11,19,12,18 X13,6,14,1 X19,4,20,5 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

- q-13/2 - q-7/2 + q-5/2 - 2q-3/2 + 2q-1/2 - 2q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + 2q-18 + 2q-16 + q-8 + 2q-6 - 1 - q2 + q8

HOMFLY-PT Polynomial:

- 2a-1z - a-1z3 + 3az + 4az3 + az5 - 3a3z - a3z3 - a5z-1 - a5z + a7z-1

Kauffman Polynomial:

2a-1z - 6a-1z3 + 5a-1z5 - a-1z7 + 7z2 - 16z4 + 11z6 - 2z8 + 2az - 6az3 + 4az7 - az9 + 8a2z2 - 24a2z4 + 17a2z6 - 3a2z8 + 2a3z - 2a3z3 - 4a3z5 + 5a3z7 - a3z9 + 3a4z2 - 8a4z4 + 6a4z6 - a4z8 + a5z-1 - 3a5z + 3a5z3 - a6 + 2a6z2 + a7z-1 - 5a7z + 5a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                     1 j = 4                   1   j = 2                 1 1   j = 0             1 2 1     j = -2           1 2 1       j = -4           1 2         j = -6       1 2 1           j = -8         1             j = -10     1 1               j = -12 1                     j = -14 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, 180]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 180]]
Out[4]=	PD[X[8, 1, 9, 2], X[3, 10, 4, 11], X[5, 14, 6, 15], X[16, 8, 17, 7], > X[22, 18, 7, 17], X[15, 13, 16, 12], X[9, 20, 10, 21], X[11, 19, 12, 18], > X[13, 6, 14, 1], X[19, 4, 20, 5], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -2, 10, -3, 9}, > {4, -1, -7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, 7, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) -(7/2) -(5/2) 2 2 3/2 5/2 -q - q + q - ---- + ------- - 2 Sqrt[q] + 2 q - q 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 2 -8 2 2 8 -1 + q + --- + --- + --- + q + -- - q + q 20 18 16 6 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 180]][a, z]
Out[8]=	5 7 3 a a 2 z 3 5 z 3 3 3 5 -(--) + -- - --- + 3 a z - 3 a z - a z - -- + 4 a z - a z + a z z z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 180]][a, z]
Out[9]=	5 7 6 a a 2 z 3 5 7 2 2 2 -a + -- + -- + --- + 2 a z + 2 a z - 3 a z - 5 a z + 7 z + 8 a z + z z a 3 4 2 6 2 6 z 3 3 3 5 3 7 3 4 > 3 a z + 2 a z - ---- - 6 a z - 2 a z + 3 a z + 5 a z - 16 z - a 5 2 4 4 4 5 z 3 5 7 5 6 2 6 4 6 > 24 a z - 8 a z + ---- - 4 a z - a z + 11 z + 17 a z + 6 a z - a 7 z 7 3 7 8 2 8 4 8 9 3 9 > -- + 4 a z + 5 a z - 2 z - 3 a z - a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 1 1 2 1 + -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 4 2 14 6 12 6 10 4 10 3 6 3 8 2 6 2 q q q t q t q t q t q t q t q t 1 1 1 t 2 2 2 2 3 4 3 6 4 > ---- + ---- + ---- + 2 t + -- + t + q t + q t + q t + q t 6 4 2 2 q t q t q t q

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