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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 7q-9/2 + 12q-7/2 - 17q-5/2 + 18q-3/2 - 19q-1/2 + 16q1/2 - 12q3/2 + 7q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - q-16 + 3q-14 - 2q-12 + 2q-10 + 3q-8 - q-6 + 6q-4 - 3q-2 + 4 - q2 - q4 + 2q6 - 3q8 + q10 - q12

HOMFLY-PT Polynomial:

a-1z-1 + 6a-1z + 9a-1z3 + 5a-1z5 + a-1z7 - 3az-1 - 14az - 24az3 - 19az5 - 7az7 - az9 + 2a3z-1 + 6a3z + 9a3z3 + 5a3z5 + a3z7

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 - 5a-3z3 + 8a-3z5 - 3a-3z7 - a-2 + 5a-2z2 - 11a-2z4 + 13a-2z6 - 5a-2z8 + a-1z-1 - 7a-1z + 16a-1z3 - 15a-1z5 + 12a-1z7 - 5a-1z9 - 3 + 14z2 - 20z4 + 17z6 - 4z8 - 2z10 + 3az-1 - 16az + 35az3 - 38az5 + 26az7 - 10az9 - 3a2 + 9a2z2 - 15a2z4 + 14a2z6 - 5a2z8 - 2a2z10 + 2a3z-1 - 7a3z + 9a3z3 - 7a3z5 + 6a3z7 - 5a3z9 - 4a4z4 + 8a4z6 - 6a4z8 + a5z - 3a5z3 + 7a5z5 - 5a5z7 - 2a6z2 + 5a6z4 - 3a6z6 - a7z + 2a7z3 - 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 r = 5 j = 10                       1 j = 8                     2   j = 6                   5 1   j = 4                 7 2     j = 2               9 5       j = 0             10 7         j = -2           9 10           j = -4         8 9             j = -6       4 9               j = -8     3 8                 j = -10   1 5                   j = -12   2                     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, Alternating, 303]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 303]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[16, 5, 17, 6], > X[18, 8, 19, 7], X[20, 14, 21, 13], X[14, 22, 15, 21], X[22, 18, 9, 17], > X[8, 20, 1, 19], X[6, 9, 7, 10], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10, 5, -9}, > {10, -1, 2, -3, 6, -7, 11, -4, 8, -5, 9, -6, 7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 7 12 17 18 19 -q + ----- - ---- + ---- - ---- + ---- - ------- + 16 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 12 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 3 2 2 3 -6 6 3 2 4 6 4 + q - q + --- - --- + --- + -- - q + -- - -- - q - q + 2 q - 14 12 10 8 4 2 q q q q q q 8 10 12 > 3 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 303]][a, z]
Out[8]=	3 3 5 1 3 a 2 a 6 z 3 9 z 3 3 3 5 z --- - --- + ---- + --- - 14 a z + 6 a z + ---- - 24 a z + 9 a z + ---- - a z z z a a a 7 5 3 5 z 7 3 7 9 > 19 a z + 5 a z + -- - 7 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 303]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 7 z 3 5 7 -3 - a - 3 a + --- + --- + ---- - --- - 16 a z - 7 a z + a z - a z + a z z z a 2 2 3 3 2 2 z 5 z 2 2 6 2 5 z 16 z 3 > 14 z - ---- + ---- + 9 a z - 2 a z - ---- + ----- + 35 a z + 4 2 3 a a a a 4 4 3 3 5 3 7 3 4 3 z 11 z 2 4 4 4 > 9 a z - 3 a z + 2 a z - 20 z + ---- - ----- - 15 a z - 4 a z + 4 2 a a 5 5 6 6 4 8 z 15 z 5 3 5 5 5 7 5 6 z > 5 a z + ---- - ----- - 38 a z - 7 a z + 7 a z - a z + 17 z - -- + 3 a 4 a a 6 7 7 13 z 2 6 4 6 6 6 3 z 12 z 7 3 7 > ----- + 14 a z + 8 a z - 3 a z - ---- + ----- + 26 a z + 6 a z - 2 3 a a a 8 9 5 7 8 5 z 2 8 4 8 5 z 9 3 9 > 5 a z - 4 z - ---- - 5 a z - 6 a z - ---- - 10 a z - 5 a z - 2 a a 10 2 10 > 2 z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 3 8 4 9 10 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 9 9 2 2 2 4 2 4 3 > ----- + ---- + ---- + 7 t + 9 q t + 5 q t + 7 q t + 2 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 5 q t + q t + 2 q t + q t

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