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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X8,9,1,10 X18,12,19,11 X20,13,21,14 X22,16,9,15 X14,7,15,8 X6,22,7,21 X4,18,5,17 X16,6,17,5 X2,19,3,20

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 8q-7/2 - 14q-5/2 + 20q-3/2 - 23q-1/2 + 22q1/2 - 21q3/2 + 15q5/2 - 10q7/2 + 5q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - q-12 + 3q-8 - 4q-6 + 4q-4 - q-2 + 1 + 4q2 - 2q4 + 6q6 - q8 + q12 - 3q14 + q16

HOMFLY-PT Polynomial:

2a-3z - a-3z3 - a-3z5 - a-1z-1 - 4a-1z - 2a-1z3 + 2a-1z5 + a-1z7 + az-1 + 2az + 3az3 + 3az5 + az7 - 2a3z3 - a3z5

Kauffman Polynomial:

- a-5z3 + 2a-5z5 - a-5z7 - 11a-4z4 + 15a-4z6 - 5a-4z8 + 4a-3z + a-3z3 - 20a-3z5 + 25a-3z7 - 8a-3z9 + 5a-2z2 - 34a-2z4 + 35a-2z6 - 2a-2z8 - 4a-2z10 - a-1z-1 + 8a-1z - a-1z3 - 37a-1z5 + 54a-1z7 - 19a-1z9 + 1 + 5z2 - 40z4 + 51z6 - 11z8 - 4z10 - az-1 + 4az - 9az3 + 3az5 + 16az7 - 11az9 - a2z2 - 10a2z4 + 23a2z6 - 14a2z8 - 4a3z3 + 14a3z5 - 12a3z7 - a4z2 + 6a4z4 - 8a4z6 + 2a5z3 - 4a5z5 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                     4   j = 8                   6 1   j = 6                 9 4     j = 4               12 6       j = 2             10 9         j = 0           13 12           j = -2         9 12             j = -4       5 11               j = -6     3 9                 j = -8   1 5                   j = -10   3                     j = -12 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, 291]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 291]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[8, 9, 1, 10], X[18, 12, 19, 11], > X[20, 13, 21, 14], X[22, 16, 9, 15], X[14, 7, 15, 8], X[6, 22, 7, 21], > X[4, 18, 5, 17], X[16, 6, 17, 5], X[2, 19, 3, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 10, -8, 7, -3}, > {3, -1, 4, -2, 5, -7, 6, -10, 9, -4, 11, -5, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 8 14 20 23 3/2 q - ---- + ---- - ---- + ---- - ------- + 22 Sqrt[q] - 21 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 15 q - 10 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 3 4 4 -2 2 4 6 8 12 1 - q + --- - q + -- - -- + -- - q + 4 q - 2 q + 6 q - q + q - 14 8 6 4 q q q q 14 16 > 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 291]][a, z]
Out[8]=	3 3 5 5 1 a 2 z 4 z z 2 z 3 3 3 z 2 z -(---) + - + --- - --- + 2 a z - -- - ---- + 3 a z - 2 a z - -- + ---- + a z z 3 a 3 a 3 a a a a 7 5 3 5 z 7 > 3 a z - a z + -- + a z a
In[9]:=	Kauffman[Link[11, Alternating, 291]][a, z]
Out[9]=	2 3 3 3 1 a 4 z 8 z 2 5 z 2 2 4 2 z z z 1 - --- - - + --- + --- + 4 a z + 5 z + ---- - a z - a z - -- + -- - -- - a z z 3 a 2 5 3 a a a a a 4 4 3 3 3 5 3 4 11 z 34 z 2 4 4 4 > 9 a z - 4 a z + 2 a z - 40 z - ----- - ----- - 10 a z + 6 a z - 4 2 a a 5 5 5 6 4 2 z 20 z 37 z 5 3 5 5 5 6 > a z + ---- - ----- - ----- + 3 a z + 14 a z - 4 a z + 51 z + 5 3 a a a 6 6 7 7 7 15 z 35 z 2 6 4 6 z 25 z 54 z 7 > ----- + ----- + 23 a z - 8 a z - -- + ----- + ----- + 16 a z - 4 2 5 3 a a a a a 8 8 9 9 3 7 8 5 z 2 z 2 8 8 z 19 z 9 > 12 a z - 11 z - ---- - ---- - 14 a z - ---- - ----- - 11 a z - 4 2 3 a a a a 10 10 4 z > 4 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 3 1 5 3 9 5 11 13 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 9 2 2 2 4 2 4 3 6 3 6 4 > ---- + 12 t + 10 q t + 9 q t + 12 q t + 6 q t + 9 q t + 4 q t + 2 q t 8 4 8 5 10 5 12 6 > 6 q t + q t + 4 q t + q t

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