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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,8,17,7 X18,11,19,12 X22,19,5,20 X20,14,21,13 X12,22,13,21 X14,17,15,18 X8,16,9,15 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 8q-11/2 - 14q-9/2 + 17q-7/2 - 20q-5/2 + 20q-3/2 - 17q-1/2 + 12q1/2 - 7q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - 2q-18 + 3q-16 + 5q-14 - 2q-12 + 4q-10 - 2q-8 - q-6 - 3q-2 + 5 - 2q2 + 3q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 + 2a-1z3 + 3az-1 + 4az + 2az3 - az5 - 4a3z-1 - 7a3z - 4a3z3 - 2a3z5 + 2a5z-1 + 3a5z + 3a5z3 - a7z

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 3a-2z6 - a-1z-1 + 3a-1z - 4a-1z3 + 7a-1z5 - 5a-1z7 - 1 + 5z2 - 7z4 + 8z6 - 6z8 - 3az-1 + 16az - 30az3 + 22az5 - 4az7 - 4az9 - 3a2 + 17a2z2 - 41a2z4 + 40a2z6 - 15a2z8 - a2z10 - 4a3z-1 + 25a3z - 52a3z3 + 37a3z5 + a3z7 - 8a3z9 - 2a4 + 12a4z2 - 35a4z4 + 42a4z6 - 15a4z8 - a4z10 - 2a5z-1 + 16a5z - 36a5z3 + 33a5z5 - 4a5z7 - 4a5z9 - a6 + a6z2 - 4a6z4 + 12a6z6 - 6a6z8 + 3a7z - 8a7z3 + 10a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     2   j = 4                   5 1   j = 2                 7 2     j = 0               10 5       j = -2             11 8         j = -4           9 9           j = -6         8 11             j = -8       6 9               j = -10     3 9                 j = -12   1 5                   j = -14   3                     j = -16 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, Alternating, 30]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 30]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 8, 17, 7], X[18, 11, 19, 12], > X[22, 19, 5, 20], X[20, 14, 21, 13], X[12, 22, 13, 21], X[14, 17, 15, 18], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -7, 6, -8, 9, -3, 8, -4, > 5, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 8 14 17 20 20 17 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 7 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 2 3 5 2 4 2 -6 3 2 6 5 - q + --- - --- + --- + --- - --- + --- - -- - q - -- - 2 q + 3 q - 20 18 16 14 12 10 8 2 q q q q q q q q 8 12 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 30]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a z 3 5 7 2 z -(---) + --- - ---- + ---- - -- + 4 a z - 7 a z + 3 a z - a z + ---- + a z z z z 3 a a 3 3 3 5 3 5 3 5 > 2 a z - 4 a z + 3 a z - a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 30]][a, z]
Out[9]=	3 5 2 4 6 1 3 a 4 a 2 a z 3 z 3 -1 - 3 a - 2 a - a - --- - --- - ---- - ---- - -- + --- + 16 a z + 25 a z + a z z z z 3 a a 2 5 7 2 2 z 2 2 4 2 6 2 8 2 > 16 a z + 3 a z + 5 z - ---- + 17 a z + 12 a z + a z - a z + 2 a 3 3 4 2 z 4 z 3 3 3 5 3 7 3 4 5 z > ---- - ---- - 30 a z - 52 a z - 36 a z - 8 a z - 7 z + ---- - 3 a 2 a a 5 5 2 4 4 4 6 4 8 4 z 7 z 5 3 5 > 41 a z - 35 a z - 4 a z + 2 a z - -- + ---- + 22 a z + 37 a z + 3 a a 6 5 5 7 5 6 3 z 2 6 4 6 6 6 > 33 a z + 10 a z + 8 z - ---- + 40 a z + 42 a z + 12 a z - 2 a 7 8 6 5 z 7 3 7 5 7 7 7 8 2 8 > a z - ---- - 4 a z + a z - 4 a z - 4 a z - 6 z - 15 a z - a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 15 a z - 6 a z - 4 a z - 8 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 3 1 5 3 9 6 9 10 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 8 11 9 9 11 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 5 t + 7 q t + 2 q t + 5 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 2 q t + q t

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